OF LUMINOUS VIBRATIONS. 591 



some intermediate value — — must be a maximum. Hence if a curve were described havinir for 

 dr " 



its equation y =f(^v), it would have a point of contrary flexure between the values of x corre- 

 sponding to y= 1 and y = 0, and would resemble the trochoid. It is also to be remarked that 

 the second term of equation (22) must be regarded as very small compared to the other terms, 

 in order that that equation may be equivalent to a linear equation in x and y, excepting where f is 

 very small. By the omission of the second term, equation (22) becomes identical with equation (20). 

 Hence, with the exception just mentioned, the curves which represent the integrals of (20) and (22) 

 coincide ; and as we found that the curve corresponding to (20) cuts the axis of abscissae in an 

 unlimited number of points, the same must be the case with the curve correspondino- to equa- 

 tion (22). But for the latter curve we have shewn that f= and — = at a point of intersection. 



dr 



Hence the motion does not extend beyond the least value of r corresponding to / = 0. 



11. The integral of equation (21) is derived from that of (12) by putting - for/ and -e fore. 

 Hence the integral of (21) in a series is, 



1 , e-r' e'r° 



7 = 1 + er' -H -— , + — + &c. 



. 3eV ige'r" 



Whence f = i - er + -t- &c (23). 



4 36 *■ •* 



This series diverges from the approximate series (19) after the second term. Let / be the least 

 value of r corresponding to f = 0. Then, 



,„ Se'l* 19e3/« 

 = I - e/2 + + jjc_ 



i 36 



eX^ 

 Hence eP is a numerical quantity. Let el' = q. Then, as we have also —^ = k, it follows that 



k = — —. Hence A; is a constant quantity for all vibrations, if the ratio - be a constant. Now 



■JT'l" I 



it may be thus argued that X and I have a constant ratio to each other. These quantities must 

 be related in some way, otherwise the motion is not defined. Let F (\, I, S) = express this 

 relation, S being the maximum condensation corresponding to /= 1. As there are no other 

 quantities concerned in this relation, and as X and I are the only linear quantities, this equation 



X X /Ic / k 



is equivalent to -- = FAS). And we have above, — = tt \/ - . Hence rr \/ - = F^ (S). But 



I * 9 7 



it has already been shewn (Art. 4) that k is independent of S. Hence F, (5) is a constant, the 

 same for all vibrations. Hence also k is the same for all vibrations. 



12. We have now found for / a particular value which satisfies the liydrodynnmlcnl conditions 



of the question, but does not admit of being definitely expressed. It can only be expressed in an 



infinite series, the terms of which do not necessarily converge. If, therefore, the pha^nomena of 



light be ex])ounded by a definite form of f, this can agree with equation (23) only under 



certain limitations. Now, by equation (is), we have a definite form of / obtained in a geiienii 



manner, without reference to the mode of disturbance. If in this e(|uation '2 A = ^ A' = },, we 



obtain, 



"e'x* 2e't/* 



/=!(!- 2e.r' + &c.) + J (I - 'H'f + ■- - &c.) 



4C2 



