## Ax Sentence Examples

- He swung the
**ax**once more, now only half a body length away. - She relinquished the
**ax**for the glass of tea. - "That's called an ice
**ax**, or piolet," Ryland answered. - Dean leaped on Shipton, clawing away at the soft snow, pummeling him like an eighth grade schoolyard brawler while Shipton, still clutching his ice
**ax**in one hand, swung at Dean, catching him on the cheek and face with the side of the solid handle. - Shipton flailed out at him with his
**ax**, missing his head by inches as Dean leaned sideways and frantically fumbled with his line to drop again. - He swung the
**ax**again. - I suppose you've got a good reason why you tried to beat the brains out of a guy holding an ice
**ax**, in the middle of the street with a bunch of people watching. - He swung his ice
**ax**into the wall in front of him, dug in the toes of his crampons and began to ascend toward Dean. - Shipton swung his ice
**ax**again, inching up closer to Dean. - Shipton's
**ax**bit the ice scarcely a foot below Dean as the man glared up at him, a snarl on his face. - The leg wound from Shipton's flailing ice
**ax**had been an eight-stitcher of no permanent consequence. - His muscular back glistened with perspiration as he swung the
**ax**, expertly splitting a chunk of wood. - This time the
**ax**sank about four inches into the wood - in another spot. - After a full minute of tugging and grunting she managed to dislodge the
**ax**from the wood. **Ax**., Green**axis**creeping on the surface of damp soil; rh., colorless rhizoids penetrating the soil; asc.**ax**., ascendingof green cells.**axes**- Such curves are given by the equation x 2 - y 2 =
**ax**4 -1bx 2 y 2 +cy 4 . - - 2 ay2'
**ax**, Ux2 ï¿½ï¿½ï¿½ a y n ay. - Taking two of the equations
**ax**+ +cz) x"' 1 +... - Other forms are n-1 n-2 2
**ax**+nbx x +n(n-i)cx x +..., 1121 2 the binomial coefficients C) being replaced by s!(e), and n 1, n-1 1 n-2 2**ax**1 +l i ox l 'x 2 + L ?cx 1 'x2+..., the special convenience of which will appear later. - As between the original and transformed quantic we have the umbral relations A1 = A1a1 d-A2a2, A2 = /21a1+/22a2, and for a second form B1 =A 1 b 1+ A 2 b 2, B 2 =/21bl +ï¿½2b2ï¿½ The original forms are
**ax**, bi, and we may regard them either as different forms or as equivalent representations of the same form. - L
**ax**2 2**ax**i l This is called the kth transvectant of f over 4); it may be conveniently denoted by (f, (15)k. - The two forms
**ax**, bx, or of, 0, may be identical; we then have the kth transvectant of a form over itself which may, or may not, vanish identically; and, in the latter case, is a covariant of the single form. - But his cry came an instant too late as Shipton plummeted past him, his ice
**ax**swinging in a rip across Dean's calf as he plummeted backward into space, and down to the rocks and churning river below. - Taking a few steps back she gripped the
**ax**half way down on the handle and slammed it down against the block of wood with a dull whack. - He lifted the
**ax**, taking aim at a new block of wood. - In the theory of surfaces we transform from one set of three rectangular
to another by the substitutions 'X=' by+ cz, Y = a'x + b'y + c'z, Z =a"x+b"y-l-c"z, where X 2+Y2+Z2 = x2+ y2+z2.**axes** - It may be written in the form n n-1 2
**ax**1 +bx1 x2 +cx 1 x 2 + ...; or in the form n n n=1 n n-2 2 +(1)bx x2+ ? - From the three equations
**ax**= alxl+ a2x2, b.= blxl+b2x2, cx = clxi+c2x2, we find by eliminating x, and x 2 the relation a x (bc)+b x (ca) +c x (ab) =0. - 32
**ax**l ay 2 ax2ay1' which, operating upon any polar, causes it to vanish. - The process of transvection is connected with the operations 12; for?k (a m b n) = (ab)kam-kbn-k, (x y x y or S 2 k (a x by) x = 4))k; so also is the polar process, for since f k m-k k k n - k k y = a x by, 4)y = bx by, if we take the k th transvectant of f i x; over 4 k, regarding y,, y 2 as the variables, (f k, 4)y) k (ab) ka x -kb k (f, 15)k; or the k th transvectant of the k th polars, in regard to y, is equal to the kth transvectant of the forms. Moreover, the kth transvectant (ab) k a m-k b: -k is derivable from the kth polar of
**ax**, viz. - Since, If F = An, 4) = By, 1 = I (Df A4) Of A?) Ab A"'^1Bz 1=, (F, Mn
**Ax**I**Ax**2**Axe**Ax1) J The First Transvectant Differs But By A Numerical Factor From The Jacobian Or Functional Determinant, Of The Two Forms. We Can Find An Expression For The First Transvectant Of (F, ï¿½) 1 Over Another Form Cp. For (M N)(F,4)), =Nf.4Y Mfy.4), And F,4, F 5.4)= (**Axby**A Y B X) A X B X 1= (Xy)(F,4))1; (F,Ct)1=F5.D' 7,(Xy)(F4)1. - Thus what have been called seminvariants are not all of them invariants for the general substitution, but are invariants for the particular substitution xl = X11 + J-s12, X 2 = 112 Again, in plane geometry, the most general equations of substitution which change from old
inclined at w to new**axes**inclined at w' =13 - a, and inclined at angles a, l3 to the old**axes****axis**of x, without change of origin, are x-sin(wa)X+sin(w -/3)Y sin w sin ' _sin**ax**y sin w a transformation of modulus sin w' sin w' The theory of invariants originated in the discussion, by George Boole, of this system so important in geometry. - Then of course (AB) = (ab) the fundamental fact which appertains to the theory of the general linear substitution; now here we have additional and equally fundamental facts; for since A i = Xa i +,ia2, A2= - ï¿½ay + X a2, AA =A?-}-A2= (X2 +M 2)(a i+ a z) =aa; A B =AjBi+A2B2= (X2 +, U2)(albi+a2b2) =ab; (XA) = X i A2 - X2 Ai = (
**Ax**i + /-Lx2) (- /-jai + Xa2) - (- / J.x i '+'**Axe**) (X a i +%Ga^2) = (X2 +, u 2) (x a - = showing that, in the present theory, a a, a b, and (xa) possess the invariant property. - (ab), aa, ab, (xa),
**ax**, xx. - To assist us in handling the symbolic products we have not only the identity (ab) cx + (bc) a x + (ca) bx =0, but also (ab) x x+ (b x) a + (
**ax**) b x = 0, (ab)a+(bc)a s +(ca)a b = 0, and many others which may be derived from these in the manner which will be familiar to students of the works of Aronhold, Clebsch and Gordan. - For the quadratic aoxi +2a i x i x 2 +a 2 x, we have (i.)
**ax**= 7/1x1+2aixix2-I-7/24, (ii.) xx=xi+xzi (ab) 2 =2(aoa2 - ai), a a = a o+712, _ (v.) (xa)**ax**= i'?- (a2 - ao)xix2 - aix2. - There is no linear covariant, since it is impossible to form a symbolic product which will contain x once and at the same time appertain to a quadratic. (v.) is the Jacobian; geometrically it denotes the bisectors of the angles between the lines
**ax**, or, as we may say, the common harmonic conjugates of the lines and the lines x x . - The ï¿½th polar of
**ax**with regard to y is n-ï¿½ a aye i.e. - Put M 1 For M, N I For N, And Multiply Through By (Ab); Then { (F, C6) } = (Ab) A X 2A Y B X 1 M N I 2 (Xy), ?) 2, = (A B)
**Ax**1B X 2B Y L I Multiply By Cp 1 And For Y L, Y2 Write C 2, C1; Then The Right Hand Side Becomes (Ab)(Bc)Am Lbn 2Cp 1 M I C P (F?) 2 M { N2 X, Of Which The First Term, Writing C P =, ,T, Is Mn 2 A B (Ab)(Bc)**Axcx**1 M 2 N 2 P 2 2222 2 2 _2 A X B X C (Bc) A C Bx M N 2 2 2 M2°N 2 N 2 M 2 2 A X (Bc) B C P C P (Ab) A B B(Ac)**Ax**Cp 2 = 2 (04) 2 1 (F,0) 2.4 (F,Y') 2 ï¿½?; And, If (F,4)) 1 = Km " 2, (F??) 1 1 M N S X X X Af A _Af A**Ax**,**Ax****Ax**Ax1 Observing That And This, On Writing C 2, C 1 For Y 11 Y 21 Becomes (Kc) K X 'T 3C X 1= (F,0 1 ', G 1; ï¿½'ï¿½1(F,O) 1 M 1=1 M 2 0`,4)) 2 0, T (Fm 2.4 (0,0 2 .F ' And Thence It Appears That The First Transvectant Of (F, (P) 1 Over 4) Is Always Expressible By Means Of Forms Of Lower Degree In The Coefficients Wherever Each Of The Forms F, 0, 4, Is Of Higher Degree Than The First In X 1, X2. - It is (f = (ab) 2 a n-2 r7 2 =Hx - =H; unsymbolically bolically it is a numerical multiple of the determinant a2 f a2f (32 f) 2ï¿½ It is also the first transvectant of the differxi
**ax****axa**x 2 ential coefficients of the form with regard to the variables, viz. - In general for a form in n variables the Hessian is 3 2 f 3 2 f a2f
**ax**i**ax**n**ax**2**ax**" ï¿½ï¿½ '**axn**and there is a remarkable theorem which states that if H =o and n=2, 3, or 4 the original form can be exhibited as a form in I, 2, 3 variables respectively. - If the forms be
**ax**, b2, cy,...