120 Mr. A. L. Selby on two 



If a, b be the mean radii of the spheres, T the period 

 of vibration, a, /3 the amplitude of vibration, 



a = a + oc sin ^r, b = b + /3 sin 2tt I y+ep 



and, supposing the pulsations small, the mean value of the 

 force is (after replacing a, b by a, b) 



lh («a 2 + 6/3 2 ) +~ (6a 2 + a/3 2 ) } . 



+ 



C" 



If the spheres are not very near, terms after that involving 

 c~ 2 can be neglected, and c tends to diminish or increase 



according as cos -^- is positive or negative — i. e, according 



as the phases of vibration of the spheres approach more 

 nearly to coincidence or opposition. 



§ 6. The above method is also applicable when the dis- 

 placement on each sphere is a Zonal Harmonic with the line 

 of centres as axis. 



Definition. — If (/, 0, 0) (x, y, z) be the coordinates of two 

 points C and P, at a distance r apart, a source at C which 



gives a potential —^jy n \ ~) a ^ ^ * s ca U e d a source of the 

 nth. order. * 



A doublet is a source of the first order, and a source of the 

 second order is derived from it in the same way as a doublet 

 from a source of order zero (hitherto called a source). A 

 source of the rth order is derived similarly from a source of 

 the r — I tli order. 



Let H (fig. 3) be the position of a doublet /j, distant/ from 

 B along BA ; and let V be the potential at (#, y, z) due to H 

 and its successive images in Y and X. 



Take a doublet \x at H ; such that BK'=f+df, and a doublet 

 — /Lt at H. 



dV 



The potential at (a, y, z) due to H7 and its images is V + —y. df. 



Therefore the potential due to a source //, of the second order 



<TV 

 and its images is — , and that due to a source p of the nth 

 aj 



d n ~ l Y 



order and its images is r * 



ft df 71 ' 1 



