482 ON PERIODIC IRROTATIONAL WAVES [419 C + cos 4^ I „ f 37a7 25a"£ 15a27 ig s ) + cos5a?|- -2() --—^-f-^-^aS + Sej ................... (13)* The constant part has no significance for our purpose, and the term in y can be made to vanish by a proper choice of g. If we use only «; none of the cosines can be made to disappear, and the value of g is g^l-tf-M-Tof ............................ (14) When -we include also /3, we can annul the term in cos 2# by making ,1K. ' ........................... (15) and with this value of $ „ 5a4 619a« But unless a is very small, regard to the term in cos 3# suggests a higher value of ft as the more favourable on the whole. With the further aid of 7 we can annul the terms both in cos 2# and in cos %x. The value of /3 is as before. _ That of 7 is given by a"/ 139a«\ ^n(l+-ir)> and with this is associated The inclusion of S and e does not alter the value of g in this order of approximation, but it allows us to annul the terms in cos 4# and cos 5x. The appropriate values are afi a7 6== 480' and the accompanying value of 7 is given by while jB remains as in (15). We now proceed to consider how far these approximations are successful, for which purpose we must choose a value for a. Prof. Burnside took a = -}. With this value the second term of ft in (15) is nearly one-third of the first (Stokes5) term, and the second term of 7 in (20) is actually larger* than the [* With the alterations specified in the footnotes on p. 481, the terms in (13) involving ct2/3?/, and (a7, a3/3) cos 3ar, become 2y . — a2/3, and cos 3z ( - ^ a7 + ~ a3/3 J. Then the highest terms in (16), (17), (18), and (20) become respectively - ^ a«, ~ ( + ^ a? \ - ^ «o, and ~ ( + ~ a the second term in (20) being now little more than half the first when a = J. W. P. S.]*" + 2/3e~^ cos 2a?