154 EVALUATION OF DEFINITE MULTIPLE INTEGRALS. 



in a series of powers and products of 



1 1 



a b " 

 and neglect all terms involving powers above the first of these 



quantities, the result will be = SO-y . 

 Consequently 



ab ... a o a 



2 - T denoting the combinations two and two of the quan- 



tities - , T . . . ; and so of the rest. For in the development 



CL O 



of (A + B + . . . ) m where m is not greater than the number of 

 quantities A, B, &c. there is always a term involving no power 

 above the first of any of these quantities, and its coefficient 

 is 1 . 2 ... m. This term is obviously the sum of the com- 

 binations m and m together of A, B, &c., and that its coefficient 

 is equal to 1.2.3 . . . m, appears from the polynomial theorem, 

 viz. 



Thus C -*- 2 -*-. , , 



-mra...(a' + a 2 )(6 2 + a 2 )... " 



where p t , p t _^ &c. are the alternate coefficients of the equation 



whose roots are a, Z>, c, '&c. (I suppose t to be the number of 

 variables in E.} In precisely the same way we should find 



77= 



Next to find the values of 



/CO -00 



I G cos a uda. and I H cos a wcfa. 



./O J o 



Let -ZT= I 84.^^ g 2>> 



3 cosawc?a 



