24 Demonstrations of certain Points in 



spherical excess, which seem to have been hitherto over- 

 looked. Thus, E being the spherical excess, the known 

 formula 



. ,^ v^ sin s sin (s— «) sin (5 — ^;) sin (5— c) 

 sm in* — + ; -, — -J -. 



— 2 cos h a cos ^ b cos ^ c 



gives for ^ E, on account of the double sign +, or of the two 

 values belonging to sin ^ E, or — sin \ E, two values, of the 

 form \ E' and 2;r— |E', and, consequently, for E the two 

 E' and 4 7r — E'. Now, the less of these, suppose E', answers 

 to the smaller of the two triangles formed by a, b, c, and the 

 other to the greater. For the excess in the former is 

 A + B + C— -tt; while in the latter, it is S^r— (A + B + C), or 

 57r— (tt + E'), or, finally, I-tt— E'. 

 (7.) In like manner, the formula 



1 + cos a + cos b + cos c 



cot i E = + o,/ ♦ • , X • / /^ • / \ f 



— ^V sm 5 sm [s—a) sm {s—b) sm [s—c) 



gives for \ E, on account of the double sign, values of the 

 form \ E', and 2 tt— i E'. So likewise, from Lhuillier's formula 



tan iE = + >/tan |^ 5 tan \ {s—a) tan i [s — b) tan \ {s — c\ 



taking the value of tan \ E first positive and then negative, 

 we find for \ E values of the form \ E^ and t:—\ E'; whence 

 E will be of the forms E' and 4 tt — E', as before. 



,. \ -n 1 r 1 , -r. cot 1 fl coti6+cosC 



(8.) From the formula cot ^ E = ^ ^^ , 



^ ' ^ sm C 



which gives the excess when two sides and the contained angle 

 are the data, we find |^ E to be of the forms \ E'and 7r + | E'; 

 whence the forms of E will be E' and 2 tt + E^ This answers 

 exactly to the two triangles mentioned in No. 6 ; since the 

 excess E' in the smaller is A + B-l-C— tt, and in the larger 

 A + TT + B + T + C— TT, or, by contraction, 2 tt + A + B + C — tt, 

 which is the same as 2 tt + E'. 



Glasgow College, Oct. 21, 1836. 



VI. Demonstrations of certain points in Fresnel's Theory of 

 Double Refraction^ deduced from the Investigations of the 

 LJyidulatory Theory isohich have recently appeared in this 

 Journal. By A Correspondent. 



To the Editors of the Philosophical Magazine and Journal, 



Gentlemen, 



¥ DO not know whether the following results are sufficiently 



■*• original to authorize an expectation that you will be able 



to afford them a place in your Journal, as, at the most, they 



