TRANSACTIONS OF SECTION A. 155 



If an electron has one quantum kinetic energj', its velocity is given by 



The difference between the two formulae for v amounts to the sixth root of v. an 

 amount which would hardly matter if only the visible spectrum were in question, 

 but is much too great when we take the X-ray region also into consideration. But 

 there is a surprising numerical agreement. If we fix our attention on two wave- 

 lengths, Ji.) that of sodium in the visible spectrum and (ii.) the wave-length 10"' cm. 

 in the X-ray region, we find that the two expressions for v give, in the case of (i.) 

 9-29 X 10' cms./sec, and 8-64 x 10' cms./sec, and in the case of (ii.) 1-68 x 10" cms./sec, 

 and 6-64x 10" cms./sec, the second value in each case being given by the quantum 

 formula. Thus, there is fair agreement for sodium, but the new formula gives only 

 one-quarter of the correct value in the X-ray region. If we make the electron fall 

 from infinity, or vary the law of density of the positive electricity, we can shift the 

 point of exact numerical agreement along the spectrum, but the power of v always 

 remains the same. 



However, the agreement, such as it is, is sufficient to make it probable, that the 

 quantum is the amount of energy acquired by a free electron in falling into a void 

 atom. 



5. On Gmiss's Theorem for Quadrature and the Approximate Evaluation 

 of definite Integrals ivith finite Limits. By Professor A E 

 Forsyth, F.E.S.— See p. 385. 



6. On certain Types of Plane Algebraic Curve. 

 By Professor Harold Hilton.^ 



The equation of the most general plane algebraic curve of degree six with 

 deficiency 1 or 0, having a triple point at which two linear branches have 

 five or six-pomt contact, while a third linear branch has ordinary contact 

 with them both, can be put in one of the forms 



{u-ay"-) (u- y-) (u-y y^) = k^xu-y^y, 

 (u-a if) {u-$ y«) (m-7 ?/-) = kiKu"; 



where a, ^8, y ai-e constants and u is written for yz-^-x'K 



If in these equations we put /t; = l and divide through by y, we get the 

 most general quintic curve of deficiency 1 or 0, whose double points all coalesce 

 at a single double point. 



In general all these double points are nodes; but one is a cusp, if one 

 or all of o, /?, y are zero. The deficiency is zero, if {&-y) (7-0) (o-;8) = 0- 

 otherwise it is unity. ' ' 



The properties of the curve may be investigated in two ways : 



(i.) The co-ordinates of any point on the curve may be expressed in terms 

 of a_ parameter t by finding the intersections of the curve with u = ty-. 



(ii.) The curve may be transformed into the cubic 



^x"-z = in-az) 0/-^:) (!/-y:) 



by the birational transformation which replaces u by y^ / z and x by x+i 

 X in the case of the second of the equations). 



7. Some unsolved Problems of Canadian Weather. 

 By Sir Frederic Stup.\rt. 



or 



loon ^'^^ P'obably be published in Rc.iidirontl (hi C'ircoJo Matematico di Palermo. 



