Mathematical Theory of Aerial Vibrations. 97 



makingy*=0, the equation (4.) may be expressed in the fol- 

 lowing form : 



Or thus : 



whence it will be seen that the terms at an infinite distance 

 either way will become insignificant on account of the factors 



(l+-) ^&c., (l--y, &c. 



Hence retaining only terms of the highest order of infinity, 

 we have, 



».. + (er2— n2)e'^-»r2«-24.(er2_w2)nV-V2«-6 + &c. =0. 



This equation is satisfied if ^r^— w^=0; or 



As no condition has been imposed on the quantity w, except 

 that it be an infinite whole number, the radius rg of the next 

 surface of no condensation will be given by the equation 



In fact, by the same process as that by which the equation 

 (tr^— n^)Q=0 was obtained, we might obtain 



(^r2-(»+l)2)Q' = o, (^r2-(« + 2)2)Q" = 0, &c., 



all the terms of Q, Q', Q", &c. being essentially positive. 

 Hence the infinite roots of the equation 



are contained in the equation 



{er'^ ^n^)[er'^-{n-\-\Y){er^-[n-\-^Y)^c. =0. 



Consequently if A be the interval between two consecutive 

 surfaces of no condensation, we have 



It is now to be remarked, that the transverse motion be- 

 PhiL Mag. S. 3. Vol. 34. No. 227. Feb. 1849. H 



