538 



Mr. J. W. L. Glaisher on Applications of 



where the mth element of the first row is or (— -1) M accord- 

 ing as m is divisible by a squared factor or is the product of 

 M simple prime factors ; and the rest of the determinant is 

 the same as in the second determinant for cr(n) in § 22. 





Expressions for <j) r ( n )) § 30. 





§ 30. Similarly we find from §§19 and 21 that 



(i-) 



M») = (-)- 1 



r, 2', f, 4", 5-, r,... 

 h 1, 1, 1, 1, 1, ... 



0, 1, 0, 1, 0, 1, ... 

 0, 0, 1, 0, 0, 1, , . . 

 0, 0, 0, 1, 0, 0, . . . 

 0, 0, 0, 0, 1, 0,... 







(ii.) 





0r(n) = (-)"-'j 1,-1, -1, 0, -1, 1, ... 





1,-2', -3', 0, -5 r , 6',... 





o, r, 0, -2", 0, -3', ... 





o, o, r, o, o, -2 r , ... 





o, o, o, r, o, o, ... 





o, o, o, o, r, o, ... 





In all the determinants the number of rows is supposed to 

 be n. 



**■ 

 The Determinant-Expressions, § 31. 



§ 31. The preceding determinant-expressions for <r r (n) &c. 

 are of course of no practical value, and they would be much 

 less convenient for purposes of calculation than the systems 

 of equations which they represent. Indeed, since in order 

 to form the determinants, it is requisite to divide not onlv n 

 but also every number less than n by all its divisors it is clear 

 that it would be much simpler to obtain the values of a(n) 

 &c. directly from their definitions. The determinant-expres- 

 sions are, however, of some theoretical interest as affording 

 definite numerical expressions for <r(n) &c. 



