Intelliyence and Miscellaneous Arlicles. 151 



The author states that he has determined by this method a hxrge 

 number of specific gravities which had been determined by older 

 methods, and had found complete agreement. He thinks that this 

 process might be applied with advantage to the determination of the 

 specific gravities of minerals, and in general of bodies whose volume 

 prevents their being submitted to the hydrostatic balance. — Comptrs 

 Rendus, December 20, 1858. 



NOTE 0\ A THEOREM IN SPHERICAL TRIGONOMETRY. 

 BY A. CAYLEY. 



1 am not aware that the following theorem has been noticed: viz., 

 in any spherical triangle, if as usual a, b, c are the sides, and A, B, C 

 the opposite angles, then 



sin b sin c-\- cos b cos c cos A = sin B sin C — cos B cos C cos a, 



sin c sin a + cos c cos a cos B = sin C sin A — cos C cos A cos b, 



sin a sin 6 + cos a cos b cos C = sin A sin B — cos A cos B cos c. 



The demonstration is very simple ; in fact we have 



sin 6 sin c+ cos b cos c cos A 



= sin b sin c (sin^ A+ cos'^ A) + cos b cos c cos A 



= sin b sin c sin- A -f- cos A (cos b cos c + sin b sin c cos A) 



= sin B sin C sin" a + cos A cos a 



= sin B sin C (1 — cos" a) + cos A cos a 



= sin B sin C + cos a (cos A — sin B sin C cos a) 



= sin B sin C — cos B cos C cos a, 



which proves the theorem. 



2 Stone Buildings, W.C., 



January 5, 1859. 



DESCRIPTION OF THE METHODS USED TO ASCERTAIN THE 

 FIGURE OF OPTICAL SURFACES. BY M. L. FOUCAULT. 



In this note the author describes three processes which he uses 

 conjointly to examine the surface of glass mirrors, in order to dis- 

 cover those parts at which the local corrections must be made, and 

 which he apj)lies afterwards. 



The first consists in placing in one of the conjugate foci of the 

 surface a luminous point, in order to observe with the microscope 

 the condition of the pencil reflected about the point of convergence ; 

 it is then seen to be decomposed into partial images, the considera- 

 tion of which furnishes sure indications as to the condition of the 

 surface itself. 



The second method is founded on the emijloyment of an object 

 with parallel sides, such as a small piece of steel wire, which is 

 placed in one of the conjugate foci, and whose image is observed at 



