IN THE NEIGHBOUrtHOOD OF A CAUSTIC. '399 



present instance 12 - m.) amounts to several integers, tlie values of the 

 successive terms decrease at first with great rapidity : yet, in all cases, 

 they increase at some part or other in hyper-geometrical proportion, and 

 finally become greater than any assignable quantity. This will be seen 

 most readily on observing the law of the terms when m = 0. 



2 1 /2\-— 2 /2\'25 



We have then r„ = --.^,: r, = \^—] . — .- : «'- = - ■——'• 



I 2 V -2.5.8 / 2\' 2.5.8.11 „ 



It is evident that these terms, however small may be the quantity 



2 ... 

 , will at some stage receive in succession new multiijliers, areater 



than ' _~~ ; that after this they will increase, and that the rapidity of 



their proportionate increase will go on continually increasing. From 

 this point then the magnitude of the terms will increase hyper-geome- 

 trically. 



The value of the integral however will be finite; and a limit for 

 the value remaining after the computation of any number of terms in 

 the series i\, v„ Ike. may be found. For, wherever we stop, the residual 



term w\\\ be of the form /' cos « . -j-^ or /", sin « . ~ : wliere i\, is the 



term last found in the series. Noav it is evident that either of these quan- 



■'fl'i) 



tities is less than I -r^ ; for the magnitudes of the quantities to be iii- 



tegrated are always smallei% except in the particular cases wlien cos // 

 or sin « = + 1 ; and their signs are constantly varying as tlie value of 



w varies : whereas the sign of -j- is always the same. Tlie residual in- 

 tegral, tiierefore, is certainly less than v,„ the last term found in the 

 series, and is ])robably much less: and tiierefore, if the last term com- 

 puted consist only of integers in tlie last place of decimals whicli we 



