Growth of a Train of Waves. 35 



looking at the modulnses shown in (209), (210), (211). 

 This allows ns to neglect co in the arguments o£ E in (205), 

 (206), and makes P and Q constant relatively to t. 



§ 157. When t is small or large, and x not so small as to 

 give preponderance to the first terms of the modulnses (209), 

 (210), we have in (205), (206), (189), (190) a full repre- 

 sentation of the whole circumstances of the wave-front, 

 extending from a?=co back to the largest value of x that 



allows preponderance of t\/ ~¥~ over <*>\/ , in the modu- 



luses, (209), (210). Let, for example, 



V£=V: • • • • <*»>■ 



This gives 



x = #L = half the wave-velocity . . (213). 

 go) 



The moving point thus defined is what in my first paper to 

 the Royal Society of Edinburgh (January 1887), " On the 

 front and Rear of a Free Procession of Waves in Deep 

 Water," I called the "Mid-Front," defined in (45) of that 

 paper, which agrees with our present (213). The folio wing- 

 passage was the conclusion of that old paper: — " The rear 

 i; of a wholly free procession of waves may be quite readily 

 " studied after the constitution of the front has been fully 

 " investigated, by superimposing an annulling surface-pressure 

 " upon the originating pressure represented by (12) above 

 ** [this is a case of (173) of our present paper], after the 

 "originating pressure has been continued so long as to 

 " produce a procession of any desired number of waves."" 

 The instruction thus given with reference to the relation 

 between front and rear has been virtually followed, though 

 with some differences of detail, in §§ 20-24 of my second 

 Royal Society paper, on the same sulject, and under the 

 same title (June 20, 1904). That second paper contained a 

 first instalment of the v " calculations and graphic repre- 

 sentations" promised in the old first; the present paper 

 contains, in figs. 35, 36, 37, 38, a further instalment of such 

 illustrations. 



§ 158. Throughout my work, §§ 96-157, 1 have had most 

 valuable assistance from Mr. George Green, not only in the 

 very long and laborious calculations and drawings, which 

 have been wholly made by him, but also in many interesting 

 and difficult questions which occurred in the fundamental 

 mathematics of the subject, 



D 2 



