96 Prof. Challis's Researches in the 



the quantities g and h being subject to the condition, 



g^ + h^= -o =4^. 

 Hence if ^=2 Vecos fl, it follows that ^=2 -/esin 9 ; so that 

 f=ct cos (2 -i/ ^ (.r cos 6 + 2/ sin fl)). 



This result is not definite, because the angle d is indetermi- 

 nate. But a result which is definite may be obtained by sa- 

 tisfying the given equation so as to embrace all possible values 

 of 9. This may be done by taking account of the analytical 

 circumstance, that the value oi f may be expressed by an 

 unlimited number of terms like that above. Let, therefore, 

 «8fl be a given indefinitely small quantity. Then we may have 



f=iXaU COS {2\/ e{x cos 9 +3/ sin fi)), 



fl having all values from 9 = to 9 = 27r. By performing the 

 summation, substituting r^ for x'^+y^i and determining a so 

 as to satisfy the condition that/=l where r = 0, the result is 



/=i-^'''+i2:2'2-32;2^:^. +&C. . . . (4.) 



It seems, therefore, that we have arrived at an expression 

 for f which involves no indeterminate quantity, and which 

 defines precisely the transverse motion. Making /= 0, the 

 resulting equation contains an unlimited number of possible 

 positive roots, and there are consequently an unlimited num- 

 ber of positions for which y*. -7^ =0, or the condensation va- 

 nishes, on any given radius. Similarly the equation resulting 



from making -^ =0, contains an unlimited number of inter- 

 ^ dr 



mediate possible positive roots ; and there are consequently 

 an unlimited number of values of/, which are radii of cylin- 

 drical surfaces situated in positions where there is no transverse 

 velocity. Since no fluid passes these surfaces, there is no 

 propagation of motion in directions transverse to the axis of 

 the ray. The intervals between the surfaces approach to a 

 certain limiting value in proportion as we recede from the 

 axis, and the maximum values of /" go on continually de- 

 creasing. The limiting value of the interval between two 

 consecutive surfaces of no condensation may be obtained as 

 follows. 



Let n be an indeterminate infinite whole number. Then 



