Mr green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. IS 



If now we effect the integrations relative to r' by means of the for- 

 mula (3), Art. 1, and reject as before those powers of the variable /•, 

 in which it is affected, with the exponent w, since these ought not enter 

 into the function Fi, the last formula will become 



(^^"F^V(3+i) 



(a) ;r-7rrj ^-rr: r'fde'diir' sin 6' Q,A„ 



2 r C ^ + ^/J-w + g^-jy x •' ^' 



and as F, ought to contain none but the odd powers of r, we may make 

 * = 2*' + l, and disregard all those terms in which s is an even number, 

 since they will necessarily vanish after all the operations have been 

 effected. Thus the only remaining terms will be of the form 



t^''^' fde'dsr' sin 9' Q,,.^, A,; 



2.T ' 



) 



where, as At and 02/+ 1 are both rational and entire functions of cos 0', 

 sin ff cos •ht', sin ff sin -sr', the remaining integrations from 6' = to 9' = tt, 

 and Tsr' = to tjt' = 2 tt, may easily be effected in the ordinary way. 



If now we follow the process employed in the preceding article, and 

 suppose To', Ti, T2, &c. are what T^, Ti, 71, &c. become when we use 

 the truncated formula («') instead of the complete one (a), we shall 

 readily get 



F, = t: + t: + t: + r/ + &c. 



In like manner, from the value of V^ before given, we get 



r," = fr'dr'd&dsr' sin &{1- ry(p{r")g' -" ; 



the integrals being taken from r' = r to r = l, from 9' = to 9' = ^, and 

 from -ar = to tsr = 2 tt. 



Expanding now g^'" as before, we have 



r;'= t;-„+t7,+ z7.+ j7, + &c. 



where 



U. = fdffd-sr' sin ■sr'Qjy'-'^dr'il-ry ^ (/*), 



