TRANSACTIONS OF SECTION A. 485 



And the more general identity is 

 sin a sin sin y sin 8 



- sin (a + X) sin (/S + X) sin (y + X) sin (8 + X) 



+ siuX sin (a + 8 + ^) sin O + 5 + X) sin (-y + 8 + X) 



- sin 8 sin X sin (8 + X)sin(a + /3 + y + 8 + 2X) = C-i) 



where a, 0, y, 8, X are any five quantities. The formula (1) is the particular case 

 of (2) obtauied by puttmg , . ^ ssx 



The formula (2) may he written 



sin (« - f) sin (« - g) sin {a - h) sin (6 - c) 



+ sin (6 - /) sin {b - g) sin (6 - /*) sin (e - «) 



+ sin(c-/)sm(c-<;)sin(e- A)sin(«-6) 



+ sin {b - c) sin (c - «) sm (« - 6) sm (re + 6 + c - / - .9 - /*) = . (ci) 

 and in this form it is in efiect due to Prof. Oayley and Mr. R. F. Scott: viz., in 

 the 'Messenger/ vol. v. p. 164, Prof. Oayley stated that a certam determinant was 

 equal to zero, if a condition, equivalent Xo a + b + c = f + g + h, was lulfiUed; 

 and in vol. viii. p. 155, Mr. R. F. Scott evaluated this determmant, without this 

 restriction, the developed result being equivalent to (3). 



16. On the Periods of the First Glass of Hyper-elliptic Integrals. 

 By William R. Roberts, M.A. 



I investi^^ate the periods of hvper-elliptic functions by a method analogous to 

 that which has been adopted by Schloemilch for the determination of the periods 

 of elliptic integrals. By this method I determine the periodicity of hyper-elliptic 

 inteo-rals without integxating the equations. 



(!)••• -yUi^^n V(=.)vrf=. V(==sH=3 = 

 ■where 



I first determine the general value of the integral / " f{,z)dz, and find it de- 

 pends on four integrals, which I call respectively, 2*, 2S, 2in, and 2i* ; and in 

 a similar manner I arrive at the general value of the integral / f{z)z"dz. The 



mode of investigation which I adopt for the determination of these integrals afibrds 

 a proof that the equation 



= = (-1)^1 

 satisfies both the transcendental equations. 



r^f(z)dz + 2/* + 2mS + 2\ia + 2/xi* = / ~ f {z)dz 



f'zy{z)dz + 21-^' + 2mS' + 2XeQ' + 2;ii^' = / " z-f {z)dz 

 By a series of transformations I proceed to show that the following equations :— 



vrr;^ =(-i)^+Vr:?5 



(2) 



