Prof. Challis on the Principles of Hydrodynamics. 87 



e being substituted for j-g. Hence at points for which f=0, 

 the velocity parallel to the axis and the condensation are con- 

 stantly equal to zero, and at points for which ;t-=0, the trans- 

 verse velocity vanishes, and the condensation is a maximum. 

 The equation /=0, it is well known, has an unlimited number 



of possible roots; and the equation -^=0, by the theory of 



equations, has a possible root intermediate to every two conse- 

 cutive roots of the other equation. Consequently there will be 

 an unlimited number of cylindrical surfaces in which the con- 

 densation is zero, and an equal number of intermediate surfaces 

 of no transverse velocity. As a preliminary step towards the 

 determination of the constant e, it is required to show that for 

 very large values of r, the intervals between the cylindrical sur- 

 faces of no condensation are equal to each other, and to ascertain 

 the value of the common interval. The following process appears 

 to suffice for this purpose. 



For the sake of convenience, substitute a^ for er^. Then the 

 wth term of the above series is 



+ 



P.2^32...(n-1)2- 



If this be greater than the (7i-f-l)th, n^ is greater than a^, or n 

 is greater than a; and if it be greater than the {n—l)th, o^ is 

 greater than [n—Vf, or n is less than a + 1. Hence the greatest 

 term is that indicated by the whole number next greater than a. 

 If a =71, the wth and (/^ + l)th terms are equal, and greater than 

 any of the others. Now it may without difficulty be shown that 

 the above series, after multiplying by 1^.2^.3^. . .w^^, may be 

 expressed as follows : 



^2n+4p-2^-4;, /^SA^^PJ ^ ^y^V 



+ 



(•4)*-('-l)'-0+*?i)' 



where the middle term of the expression consists of the nth and 

 (71 + l)th terms of the series; the first term of the expression 

 gives every pair of terms preceding the «th by substituting for jo 



