THE THEORY OF TERRESTRIAL MAGNETISM. 433 



In the same way it is shewn that the separate formulae for 



f 1 (Xtfdp, I' (Y.Jdp. and f 1 (Ztfdn, 



J -i J -i J-i 



when added together give the same result as the formula for 



+ ( 7,')' + ( 



-i 

 and that this result is 



Similarly the formulae of Art. 15 (p. 425) give 



s. 



21. The important function I X"X"dft + YYdp. was first proved 



)-\ J-i 



to vanish in all cases by a different method from that given above. 



It was shewn that the function contains the factor 



f , o f 1 n, m 1 



\ m + 1m- -+ T- 



L/, + m 1 ( 



. . 

 - 1) (n. + m 2) 



(rt.-m-l)(n 1 -w-2) 



" 



Also it was proved by induction that 



c\ \ *" / i V / r*\ / r\ "i Ot-C- 



x-l (x-l)(x-2)^ (x-l)(x-2)(x-3Y 



(-l)(a;-2) ...(x-r) ~ x-y' 

 For when ?/ = 1 it becomes - - , 



when y - 2 it becomes - + = ^" - = ^^ 



X-^l I >l/ L I \JL ) \JU 1 1 IC ^/ t</ ^ 



Assume the same law to hold for y l = y 1 and prove it to be true 

 for y thus 



A. n. 55 



