In considering the relative brightnesses of the different spectra, it is therefore sufficient to attend merely to the principal directions, provided that the whole deviation be not so great that its cosine differs considerably from unity.
Thus if x= log x =1 - where the path of integration in the plane of the complex variable t is any curve which does not pass through the origin; but now log x is not a uniform function, that is to say, if x describes a closed curve it does not follow that log x also describes a closed curve: in fact we have log (E +in) = log,/ Q 2 +7)2) t i (a + 2 n 7r), where a is the numerically least angle whose cosine and sine are (2 +n 2) and 71/,i (t 2 +n 2), and n denotes any integer.
As far as the circlesquaring functions are concerned, it would seem that Gregory was the first (in 1670) to make known the series for the arc in terms of the tangent, the series for the tangent in terms of the arc, and the secant in terms of the arc; and in 1669 Newton showed to Isaac Barrow a little treatise in manuscript containing the series for the arc in terms of the sine, for the sine in terms of the arc, and for the cosine in terms of the arc. These discoveries 1 See Euler, ” Annotationes in locum quendam Cartesii," in Nov.
Since dr/ds is the cosine of the angle between the directions of r and ~s.
C2P2=IC2cos obliquity (The obliquity which is found to answer best in practice is about 143/4; its cosine is about i3/4, and its sine about 3/4.
Consequently, one of the forms suitable for the teeth of wheels is the involute of a circle; and the obliquity of the action of such teeth is the angle whose cosine is the ratio of the radius of their base-circle to that of the pitch-circle of the wheel.
If six films of the same liquid meet in a point the corresponding tetrahedron is a regular tetrahedron, and each film, where it meets the others, has an angle whose cosine is - i.
For mercury in a glass tube the angle of contact is 128° 52', the cosine of which is negative.
After Rankine, a French engineer, Bouvier, gave the ratio of the maximum stress in a dam to the maximum vertical stress as 1 to the cosine squared of the angle between the vertical and the resultant which, in dams of the usual form, is about as 13 is to 9.
On examining the diagram it will be observed that the maximum compressive stresses are parallel to and near to the down stream face of the section, which values are approximately equal to the maximum value of the vertical stress determined by the law of uniformly varying stress divided by the cosine squared of the angle between the vertical and the resultant.