Prof. Cayley on Lobatschewsky's Imaginary Geometry. 231 



assembled at Bath, before my experiments on calorescence had 

 commenced. 



If I have not waited longer before publishing the results of my 

 experiments, as I once proposed to do, it is because by his un- 

 warranted and unprovoked attacks, Dr. Akin has forfeited all 

 claim to the position in which I had voluntarily, but unwisely, 

 placed him. 



Royal Institution, February. 



[The Editors have received a letter from Dr. Akin, in which he 

 begs to enter a public and formal protest against the note appended 

 to his communication in the last Number of the Magazine by 

 W. F., and claims to. have the last word in this discussion. As Dr. 

 Akin commenced the controversy in the December Number, and in 

 that of February has had every opportunity of placing all the evi- 

 dence he deemed necessary before our readers, we cannot admit 

 the justice of his claim, and abide by the decision already given. 

 — The Editors.] 



XXXIII. — Note on Lobatschewsky's Imaginary Geometry. 

 By A. Cayley, Esq.* 



VIJRITING down the equations 





1 cos A -f- cos B cos C 

 — , = cos a = 



sin B sin C 

 1 7 cos B + cos C cos A 



= COS = : 7^—. ; 3 



cos b l sin C sin A 



1 cos C + cos A cos B 



= cos c 



cos d sin A sin B 



where A, B, C are real positive angles each <^-7r : first, if 

 A + B + C>7r, then a, b, c are real positive angles each less than \ir 

 (this is in fact the case of a real acute-angled spherical triangle), 

 but a!, V, d are pure imaginaries of the foimjfi, q'i, r f i (where 

 p ] ,q* ,r ] are real positive quantities ; and secondly, if A -f B + C<7r, 

 then a, b, c are pure imaginaries of the form pi, qi, ri (where 

 p, q, r are real positive quantities), but a', V , c' are real positive 

 angles each less than \tt. Hence assuming A-f-B + C-^7r, and 

 writing ai, bi, ci in place of a, b, c, the system is 



1 . cos A + cos B cos C 

 . = cos ai = 



sin B sin C 



1 , . cos B + cos C cos A 



= cos bi — 



cos V sin C sin A 



* Communicated by the Author. 



