312 Mr. J. J. Sylvester: Meditation on the 



overlook so obvious and important an extension of the principle 

 and its applications, I find hard to realize; and my wonder 

 is even greater that I should not have been anticipated twenty 

 years ago, than that I should have been anticipated so recently. 

 But the algebraical theory to which this extension points the 

 way is replete with interest of a far higher order than its appli- 

 cations to practice ; for plainly the derived approximate frac- 

 tions, however sufficient for the purposes of computation, are 



proaches indefinitely near to unity, the constant left undetermined being 



known to be less than log 2+ q* 



Nay, the demonstration may be made absolutely rigid if we set about to 

 find an inferior limit. To this end make 



we shall find without difficulty 



i i l-f-y-6 y +b 



I 



*dx=jr log y _ 1+6 • ;pj, where y = Vl +b\ 

 and consequently we shall obtain as an inferior limit to F(c) the expression 



yiog^n+6 '-yip 



2 



which approaches indefinitely near to logr- as c approaches indefinitely 



near to unity. It is thus seen that Legendre's F(c), when c is indefinitely 



2 2 77 



near to 1, lies between log r and log ^ -f qJ the arithmetical mean between 



'2. TV 1 



these limits is log^ + j, i. e. log ^ +1*4785, differing by only *0923 from 



the true value log^ + log 4. Of course, when the form of F(c) in the case 



supposed is known, viz. log ^ -f C, there is no difficulty in determining C 



(as may be seen in Verhuist's Traits de Fonctions Elliptiques) ; but the 

 process above given of throwing the general value of F(c) between limits, 

 is, I believe, by far the easiest and most natural method of obtaining this 

 form. The limits themselves, it should be noticed, h ave vir tually been 

 found by the method, simple to naivete, of writing Vl — cV= V^+g 2 , 



where »= V I — x 2 and q= bx, and then substituting for -7 -7- * —7— as an 



p+q .... V^+2 3 p+q 



inferior, and a , a as a superior limit in the quantity to be integrated. 



Closer and calculable limits ad libitum to the integral may be arrived at by 



substituting for / 8 a one or the other of the two following rational 



functions of p, q, according as we wish to obtain an inferior or superior 



