of Attractive and Repulsive Forces. 175 



/= 1 - er ! + p-g, - p^3- 2 + Ac 



In these expressions e is a constant introduced by fulfilling the 

 condition that (f> is a function of 2 and t and / a function of x 

 and y ; c is a constant altogether arbitrary ; m determines the 

 intensity and X the extent of the vibrations ; and r is the dis- 

 tance of any point from the axis. It is proved also (Prop. 



XIV. pages 214-224) that the factor (l + %f has the fixed 



numerical value 1-2106. The exact determination for deter- 

 mining </> is found to be (p. 203) 



4*^-^+^ + 2^. ^ + ^^ = (a) 

 ±ae<p a ^ ■+.#+*& dzdt + dz ^ dz * ~»' ' W 



If, for the sake of shortness, k be put for (lH g" )\ ff° r 



z — iced + c. and q for — - , this equation gives by successive ap- 



proximate integrations (p. 206), 



d(j) . v Am 2 Bm 3 . ' ,, - 



-~ -=m sin qc cos 2qc — sin 6qc + &c, 



dz a ar x 



A , . , , 2a: , ^ ■- 3(7* 2 +l) 



A being put tor 777^ — rr> an d Jt> tor v t/ ., — -—• 

 & x o(/r — 1) oz(/r — iy 



After making the more general assumption (pp. 226 & 227) 

 that 



, mf „ m 2 Ag . a „ , m*Bh v 



*= - 7 cos « ? ~ v sm 2 ? ?+ 3^? cos 3 ? ? > 



I found, on substituting this value in the general differential 

 equation of which yjr is the principal variable, that the factors 

 g and h, as well as /, are determinable as functions of r, the 

 results being /= 1 — • er 2 + &c. (as before), 



2>k 2 — 1 

 g = 1 — er 2 j — e 2 ? A + &c, h= 1 — er 2 + &c. 



Then, as the differential coefficients -~ and -y£ give for any 



point the velocities parallel and transverse to the axis, the con- 

 densation at the same point is obtainable from the equation 



