When substances in solution are dealt with, Thomsen indicates their state by affixing **Aq** to their symbols.

Thus [NaOH **Aq**, HNO 3 **Aq**] =13680 cal.

Thus the equation Cl 2 -1-2KI, **Aq**=2KC1, **Aq**+12+52400 cal., or (C12) +2KI, **Aq** =2KC1, **Aq**+[12]-I-52400 cal., would express that when gaseous chlorine acts on a solution of potassium iodide, with separation of solid iodine, 52400 calories are evolved.

When C vanishes j has the form j = pxg x, and (f,j) 3 = (ap) 2 (**aq**)ax = o.

Hence, from the identity ax (pq) = px (**aq**) -qx (ap), we obtain (pet' = (**aq**) 5px - 5 (ap) (**aq**) 4 pxg x - (ap) 5 gi, the required canonical form.

A similar expression can be found for Q'P - Q"A; and thus, if Q' A =v, Q' AO = where v =a cos (0", we get - - -**AQ**' = a sin w (sin 4 -sink") - - 8a sin 4 w(sin cktan 4 + sin 'tan cl)').

The grating at A and the eye-piece at 0 are rigidly attached to a bar AO, whose ends rest on carriages, moving on rails OQ, **AQ** at right angles to each other.

A =p n, b= **aq**, c- 2n bq, and so on.

From equations (i), we find that **aq** _ o a pi qi -, so that the extension of the new range is seen to be dp i dq i ...

The generalized formula is fxo u¢ (x)dx = **Aq**(xm) - where 0, A 2, ...

If we suppose alb to be converted into a continued fraction and p/q to be the penultimate convergent, we have **aq-bp**= +1 or -1, according as the number of convergents is even or odd, which we can take them to be as we please.

If we take **aq-bp**= +1 we have a general solution in integers of ax+by=c, viz.

X = cq - bt, y = at -cp; if we take **aq-bp**= - I, we have x=bt-cq, y=cp-at.

Let APB be a semicircle, BT the tangent at B, and APT a line cutting the circle in and BT at T; take a point Q on AT so that **AQ** always equals PT; then the locus of Q is the cissoid.