436 Prof. Sylvester on a Memoria Technica 



In the hope that these remarks may induce some of your 

 readers to direct their thoughts to the elucidation of an im- 

 portant point in physics, 



I remain, 



Yours faithfully, 



Charles Brooke. 



LX. Note on a Memoria Technica for Delambre's, commonly 

 called Gauss's Theorems. By Professor Sylvester, F.R.S* 



THE most subtle reagents employed in spherical analysis and 

 transformation are the following four admirable for- 

 mulae, "commonly ascribed to Gauss, but in reality due to 

 Delambre"t: — 



c A + B . C a+h 

 cos jr cos — - — = sin — cos — ~ ) 



& & £1 & 



c . A + B C a-b 



cos — sin — - — = cos — cos — 5— > 



. c A— B . C sin (a + b) 

 sm-cos— ^— = sm^ — -^ > 



. c . A— B C sin (a — b) 



sm _ sin __ =cos _ — _ — 



Pour out of the six binary combinations of these four equations 

 give by simple division Napier's Analogies, a term which seems 

 almost equally appropriate to designate Delambre's formula. 

 It need hardly be remarked that whilst Napier's analogies may 

 be immediately deduced from Delambre's formulae, the converse 

 is not true. 



If we call the products on the left-hand side of the equation 

 P, Q, R, S, and their polar reciprocals P', Q', B/, S', it is worthy 

 of notice that the formula become 



P=-P', Q = E/, R = Q', S=-S'. 



The formula may be expressed collectively by the easily re- 

 membered disjunctive elective equation 



cos c cos A+B cos C cos a±b 



• 2 - 9^ =■ ■ 2 ■ 2 " 



sin sm " sm * sin 



* Communicated by the Author. 



t Todhunter's 'Spherical Trigonometry/ p. 27. See also Davies's 

 edition of Hutton's Course, vol. ii, p. 37. 



