4 Lord Kelvin on an Initiational Form. 



and in all o£ §§ 1-31, this notation cf> and — £ was consistently 

 used, with — f to denote, when positive, upward displace- 

 ment of the water (represented by upward ordinates in the 

 drawings). In the two curves of § 4, fig. 1, that which has 

 its maximum over represents (137), for £ = 0. The other 

 curve of fig. 1, with positive and negative ordinates on the 

 two sides of 0, represents (137), with — {RD} instead of 

 {RS}. The symbols {RS} and {RD} were introduced in §3 

 above ; {RS[ to denote a realisation by taking half the sum 

 of what is written after it with +t, and {RD} to denote a 



realisation by taking — of the formula written after it 



1 , 



minus tt of the same formula with + 1 changed into — t. 



2i ° 



A new curve in which the ordinates are numerically equal 



to •— ■ >tv t~ of the ordinates of the second of the old curves 



^/2 dx 



of fig. 1, is now given in the accompanying diagram, fig. 33 ; 

 and close above it the first of the old curves of fig. 1 is 

 reproduced, with ordinates reduced in the ratio 2^/2 to 1, 

 for the sake of comparison with the new curve. This new 

 curve represents the more convenient initiational form re- 

 ferred to in the title of the present paper. 



Its equation, found by taking £ = in (139) or in (111) 

 [most easily from the imaginary form of (139)], is as follows: 



t (.,, ,,0) = ^^±^(2c- p) . . (138). 



§ 101. The original derivation of the new particular 

 solution, (which we shall call ^,) from the primary (136), 

 as indicated in § 100, is shown by the following formula : 



*(,,,, 0HKD } £^e^> 



d -1 



^/{z + ix) 



gf\(139), 



1 f (ql-x 3 \ lgf- /gt-x 5 \ "> -tf 



= vn cos lv "VQ- 2 P cos (v ~2 x )r 4f " 



where p = \/ (~ 2 + .i' 2 ) , and % =■ tan _1 [xjz) . 



An equivalent formula for the same derivation, which will be 

 found more convenient in §§ 135-157 below, is as follows : 



