IN THE NEIGHBOURHOOD OF A CAUSTIC. 397 



each series of - {w^ - m.w) was formed by applying - - m.iv to the first 

 term of the series of ^ ic^ -. and the first difference was formed, all through 

 the series, by applying -^ m .Sw to the corresponding 1st difference of 

 lif. The last terms of all the series were compared together, as a 



TT 



check. Then for every term the arc less than ^ was taken whose sine 

 or cosine (with proper sign) represented cos ^ (w' - m . w). The natural 



til 



numbers were taken to the 7th decimal place, and differenced, as has 

 been mentioned. The number of arguments thus computed is 5166; 

 and the number of natural terms for the summation is the same; and 

 the whole of these liave been differenced on paper to the third order, and 

 mentally to the fourth order. 



The integration as far as lo = 2-00 being thus completed, and with 

 the utmost accuracy, the next step was to compute the integral from 



u> = 2-00 to w = infinity. Let ?i = - (w^ — m . tv) : the problem is now 



to find L cos u. If we make -r- = v, this integral 

 ' an ^ 



dw 



. dit . . . fiv . ^ dv . du 



— Lv . cos 11 -r- =v sm « — L sm u -f— = v sm u — ft -r- . sm Ji -r- , 

 div -"^ dw ■' div die 



where the last term may be integrated by parts as before. Proceeding 

 with this operation, and putting 

 v„ = V, 



dv 

 dw 

 _ d / dv\ 

 dw \ dwi ' 



_ _^ r ~ ( ^^\i 



dw L dtv \ dwj J ' 

 d i d r d I dv\-\\ 



3 E 2 



