OF ARBITRARY CONSTANTS, &c. 121 



Now SpX - X^s a maximum for X = pk Let \ = p^ + ^; then 



3pX-X^=2pi-3pi^'-^', 

 and our integral becomes 



e'>'*f''^e-'<'*^-^d^. 

 Put r= 3'ip~i^ ; then the integral becomes 





Let now p become infinite ; then the last integral becomes / e'^^d^ or 7ri For though 



the index - P- S"^^"^^-' becomes positive for a sufficiently large negative value of ^, that 

 value lies far beyond the limits of integration, within which in fact the index continually 

 decreases with ^, having at the inferior limit the value — 2pi. Hence then for 6 = 0, and for 

 very large values of p, we have ultimately 



u = 3-i',r-'{E + F)p-ie^''*. 



Next let = — . In this case ax = - p, and we get for the leading part of « 



3 



aETe-'-'+'i'^dX, 

 Jo 



I (29) 



which when p is very large becomes, as before, 



S-^7riaEp-ie'i'\ 



27r 

 Comparing the leading terms of u both for = — and for 6=0, we find, observing 



that a = e^ ' 



C=\/^3-iniE, 

 D=S-iTri{E + F) 

 Eliminating E, F between (28) and (29) we have finally 



^ = ,r-r(l){ C4-e-^^2)f,l _ _ ^^^^ 



5= 37r-^r(f){-C + e?^^'Z>^.| , 



20. As a last example of the principles of this paper, let us take the diflFerential equation 



<f M 1 du 



-— + - — -M = (31) 



dx x dx 



The complete integral of this equation in series according to ascending powers of a? involves a 

 logarithm. If the arbitrary constant multiplying the logarithm be equated to zero we shall 

 obtain an integral with only one arbitrary constant. This integral, or rather what it becomes 

 Vol. X. Part I. 16 



