for Delambre's, commonly called Gauss's Theorems. 437 



The number of products on each side of the equation, if all the 

 combinations of trigonometric affection and algebraical sign are 

 exhausted, is 2 3 or eight. Out of each 8, 4 only are to be pre- 

 served and colligated each with each. Thus the number of 

 systems capable of formation is 



( i '. I '. 3 '. I T (1 • 2 • 3 • 4) =24 x 70 *= U7m > 



of which one only is valid. This accounts a priori for the diffi- 

 culty of recollecting these formulae, a difficulty often complained 

 of and still oftener felt, and which is one reason of their being 

 comparatively little used by junior students. Two observa- 

 tions easily retained in the memory will serve, I think, in a great 

 degree to remove this difficulty. 



Rule 1. On opposite sides of any one equation the trigonome- 

 tric affections of the angles are contrary, and those of the sides 

 similar. 



Rule 2. The trigonometric affection of the uniliteral factor of 

 each product governs the algebraic sign of the biliteral factor, in the 

 following manner : — 



Comparing products which lie on the same side of the equations, 

 like and unlike affections go with like and unlike signs ; compa- 

 ring those which lie on opposite sides of the equations, unlike and 

 like affections go with like and unlike signs. 



These two rules are not quite sufficient in themselves \ for they 

 would be satisfied not only by the four true equations, but also 

 by the four following false ones : — 



c A-B . C a— I 



cos— . cos — - — =sm- cos 



2 2 2 ' 2\ 



c , . A-B C a + b 



cos-- . sin — jz — = cos—, cos —^r- j 



Z Z \Z i-i 



c 



A + B C . (a-b) 



sin- . cos— - — = sin — . sin h 



4 A A' A 



. c . A + B C . (a + b) 



sin — sm — -: — = cos ^ sm -^— - — - • 



A & & d 



To make the system of rules complete so as to exclude a priori 

 the construction of the four false deductions, it is necessary and 

 sufficient to bear in mind that, on the left-hand side of the 

 equation, the cosine-affection of the uniliteral term is associated 

 with the plus sign in the biliteral one*. 



* Rule 2, with the addition to ii", may be easily retained in the memory by 



