THE EQUILIBRIUM OF FLUIDS. 173 



the limits of the integrals being the same as before. But if we 

 make & + <, there will result d& = d$ 9 and 



cos iO'= cos iO cos i(f> sin id sin /<, 



and hence the double integral here given by observing that the 

 term multiplied sin fy vanishes when the integration relative to 

 (f> is effected, becomes 



A n Ad) ' j ' / '2 2V d> cos 



cos i0f*A (l} r dr (I - r 2 



If now we write V t (i} for that portion of V which is due to the 

 term a t (i] . r in the coefficient A (i) we shall have 



. cos tflV"** 1 dr' 1 - / 



But by well known methods we readily get 



d(> cos i(> 



r 



J o 



- 2rr' cos < + / 



j a-n-i^oo ~....- 



2 " 



2 . 4 ...... 2f 2 . 4 



when / > r, and when r' < r, the same expression will still be 

 correct, provided we change r into r and reciprocally. 



This value being substituted in that of V t (i} we shall readily 

 have by following the processes before explained, (Art. 1 and 2), 



3 



/2/8+5+2t-2A 



