42 Prof. Challis on Hydrodynamics. 



is not allowable to take into account this second term, all the 

 reasoning subsequent to the determination of the forms of the 

 component vibrations having excluded terms of the order of the 

 square of the velocity, whether the velocity be component or 

 coaipounded. But it is legitimate to regard Aor l as representing 

 the part of the total condensation which corresponds to the sum 

 of terms which contain m 2 as a factor and do not change sign, it 

 having been proved that the law of coexistence of vibrations holds 

 good notwithstanding that the expressions for the component 

 condensations contain such terms. 



This point being settled, I proceed to the determination of the 

 constants m x and m\. The mathematical reasoning which this 

 investigation will require is simply additional to that contained 

 in Part IT. It will be supposed that the waves are incident on 

 the sphere, regarded as fixed, in the negative direction of propa- 

 gation, and that they are defined by the equations 



V= — A:«S = msin — - (r cos 6 + icat + c ) . 

 x 



At distances r from the centre of the sphere very small com- 

 pared with X, and very large compared with c the sphere's radius, 

 the disturbance of the incident waves by the presence of the 

 sphere is conceived to be of insensible magnitude. In this case 



the arc is always very small within the extent of the 



disturbance, and it will suffice to expand the sine and cosine of 



r 2 . . 2tt 



it to terms inclusive of -5-. Accordingly, putting q for -— , the 



A, A, 



value of V may be put under the form 

 V = — fcaS=m$inq (tcat-{-c ) 



qr cos qc cos 6 — ^-sin ^c cos 2 6 \ 



— insinq/cati qr sin qc Q cos 6 + ^— — cos qc cos 2 6 I. 



The terms involving qr are so small that their effects may be 

 considered independently of each other, and also independently 

 of the still smaller terms involving q 2 r 2 . Taking, in the first 

 place, the term containing qr sin qicat, and substituting —Kaa x 

 for m sin q (jcat + c ), we have 



S — <T 1 = — ~ sin qc sin qicat cos 6. 



KCL 



To this case the first of the two particular integrals treated of in 

 the October Number is applicable, and we may at once adopt 

 the expression for a there obtained, which, when the arbitrary 



