54 6 SCIENCE PROGRESS 



the other. Watson (ibid. 49-55) discusses the limits of applica- 

 bility of Kelvin's principle, giving a formal analytical proof of 

 a theorem slightly more general than Kelvin's theorem on an 

 asymptotic formula for a certain integral. Watson {ibid. 

 96-110) obtains" aproximate formulae which exhibit the be- 

 haviour of a Bessel function when its order is large, right through 

 the transition stages between the domains of validity of the 

 three known formulae for the dominant terms of the asymptotic 

 expansions of the function for various ratios of the argument 

 to the order of the function. His formulae are more exact 

 forms of some approximations obtained by Nicholson in 

 1910. 



F. J. W. Whipple (Proc. Lond. Math. Soc. 1917, 16, 301-14) 

 obtains a certain relation between Legendre's P and Q func- 

 tions with parameters cosh a and coth a, which explains the 

 symmetry of various parts of the theory of Legendre's func- 

 tions. Its geometrical interpretation is found in the applica- 

 tions of inversion to potential problems connected with toroidal 

 co-ordinates. 



M. Bocher (Trans. Amer. Math. Soc. 191 7, 18, 519-21) gives 

 a note supplementary to his paper of 1901 on certain pairs of 

 transcendental functions whose roots separate each other, and 

 which contains a further general theorem discovered since the 

 former paper was published. 



F. H. Safford (Bull. Amer. Math. Soc. 191 7, 24, 74-6) obtains 

 a new method of reducing the general elliptic element to the 

 Weierstrassian form, which is allied to the formula published 

 by G. G. A. Biermann in 1865 as derived from Weierstrass's 

 lectures. 



Dunham Jackson (ibid. 77-82) proves, by a method different 

 from that of Frechet in his paper (1906) on the functional cal- 

 culus, that a suitable change of parameters in the representa- 

 tion of an arbitrary continuous curve, x = f(t), y = <f>(t), will 

 eliminate the intervals where / and $ are constant together. 



E. W. Chittenden (Trans. Amer. Math. Soc. 191 7, 18, 161-6) 

 shows the correctness of the equivalence conjectured by Frechet 

 (1910) of his " ecart " and " voisinage." 



F. L. Hitchcock (Proc. Roy. Soc. Edinburgh, 191 7, 37, 

 250-55) examines and classifies the various cases in which we 

 can obtain a solution of Tait's functional equation <£' = «, 

 where o> is a given linear vector function and (f> is to be found. 



