578 MR THOMAS STEPHENS DAVIES ON 



SECTION I. LEMMAS TO BE USED IN THE DISCUSSION. 



LEMMA I. 



To expand (r 2 2r r^ cos w + r 1 2 ) in multiple cosines, n being a positive integer. 



Put 2 cos w = u + - : then, 

 u 



1 ^ \ at 



Expand both these by the binomial theorem, writing 4, t v t?, . . . t n _i, t n for 

 the co-efficients of the first, second, third, . . ., nth, (n + 1 )th terms of the expan- 

 sion (1 + 1)". We thus have the two series. 



-* -V - t z r"^ -L- + ^r"-* -L - .... 



IT M* M* 



The following simple arrangement of the order of multiplication will enable us 



(r \ * 

 - ) simultaneously, and thence the value of cos (* w) for the 



successive values of k from to n. 



1. Multiply each term of the first series by that one of the second which 

 stands immediately beneath it : the sum of these products is that term of the 

 expansion which is clear of cosines, or that in which k=0. 



2. Multiply each term of the first series by that one of the second which 



stands immediately to the right of it, which will be the co-efficient of - : then mul- 

 tiply each term of the second series by that one of the first which stands imme- 

 diately to the right of it, which gives the co-efficients of u. These co-efficients will 



evidently be identical. Whence we shall have the compound co-efficient of u + - 



or of 2 cos w. 



3. Multiply cross- ways as before, the multipliers being in this case two steps 

 to the right, instead of one, as in the preceding case : the result will be the com- 

 pound co-efficient of w 2 + -5- or of 2 cos 2 w. 



u' 



4. Taking, in like manner, the multipliers three steps to the right of the terms 

 multiplied, we shall obtain the compound co-efficient of u 3 + or of 2 cos 3 co. 



Proceeding thus, we shall have every term of the first series multiplied by 

 every term of the second, without any repetition of the same factors : and the 

 required expansion will be found to be (R 2 denoting the vinculated trinomial), 



