146 Prof. Challis on the Mathematical Theory of 



deprived of this term. But to this proceeding there lies the objec- 



d 2 f . 

 tion, that where the middle term is a maximum. -^ is equal to 



df 



■ij however large r may be. Thus, since at very remote distances 



from the axis there may be values of the first and second terms 

 comparable with each other, the latter term cannot be legiti- 

 mately omitted for large values of r. 



Substituting now for convenience x 2 for er 2 , the general or 

 ?ith term of the right-hand side of the equation (12) is, irre- 

 spective of the sign, 



x 2n ~~ 2 



l 2 2 9 .3 2 ...(n-l) i 



If this be greater than the n-f 1th term, x 2 is less than n 2 ; and 

 if it be greater than the n — lth term, x 2 is greater than (n — l) 2 . 

 If x <2 =n 2 , the nth. and ft + 1th terms are equal to each other, 

 and greater than any preceding or following term. This being 

 premised, let us next give to the equation (12) the form 



( -t\n-p + l 



l 2 . 2 2 . 3 2 . . . n 2 



[_^.( 1 + i)- 2 ( 1 + ?)- 2 ...( 1+ £)-> + ^ 



in which n may be supposed to be any integer. By putting for 

 p in the first term within the brackets the values n — 1, n — 2, 

 ... 2, 1, 0, all the terms to the ftth inclusive are obtained ; and 

 by putting for p in the other term the values 0, 1, 2, . . . ad infi- 

 nitum, all the remaining terms are obtained. Let us now sup- 

 pose that x is very large, and that it has the particular value n, 

 which makes the two largest terms cancel each other. Then the 

 largest terms will be very distant from the first term. The ex- 

 presion for / above shows that the same value n x of x which 

 makes the two largest terms disappear, cancels also all terms of 

 the next inferior order with respect to n v so that for this value 

 the equation becomes, as far as regards the terms which can be 

 paired together, of the form 



f=An l 2n *- 2 + Bn l 2n >- 3 + &c, 



the first term of which is removed by two degrees from the 

 greatest terms. With respect to the terms which cannot be 

 paired, it is evident that they will be comparatively very small 

 on account of their distance from the largest terms, and the small 



