TRANSACTIONS OF SECTION A, 555 



the results is not easily seen, and so we may use the equivalent form (given on 

 p. 21G of my paper) 



which is Charller's form. It is perhaps worth while to remark that Jordan's 

 method ' can be applied in this case, and without the use of imaginary quau- 

 tities. 



10. The Puiseux Diagram and Differential Equations.'^ 

 By E. W. H. T. Hudson, B.A, Felloio of St. John's College, Cambridge. 



The paper is concerned with the approximate solution of ordinary differential 

 equation.^ in the neighbourhood of singular points, and commences with a brief 

 description of the method of using a diagram of unit points (squared pa])er) 

 similar to that introduced by Puiseux for the case of algebraic functions. This 

 method, which was first applied to differential equations by Briot and Bouquet, 

 and extended by Fine, is shown to be capable of supplying information as to the 

 existence of infinitudes of non-regular integrals which are usually obtained by 

 purely analytical processes. The essential thing to notice is that a first approxi- 

 mation to a solution may be obtained, not only ixoma. side of the ' polygon,' but also 

 from a corner, provided that the corner arise as a marked point from two or more 

 terms in the differential equation, and two inequalities be satisfied, expressing a cer- 

 tain geometrical condition. The case of a differential equation of the first order 

 and a point on the discriminant locus at which the integral curves have not a cusp 

 is a good example, and shows the existence of a namd may be predicted from an 

 inspection of the diagram. The case of solutions in series which at some stage 

 introduce logarithms is shown also to depend on corner points arising from more 

 than one term. 



11. The Fourier Problem of the Steady Tem^yerattires in a thin Rod. 



By James W. Peck. 



The solution d = V exp | - .r ^'^FA is considered from the point of view of the 



isothermals and tubes of flow. The result so got appeai-s to contradict the initial 

 hypothesis of lateral radiation ; and it is pointed out that tbe difficulty cannot be 

 evaded by considering the radiation negligible, for this nullifies the initially chosen 

 equation of heat-flow. Explanation is found in the approximate nature of the 

 solution, and two necessary conditions of the approximation are worked out as 

 follows: defining the ratio e : /,; as the thermal length modulus (L) — also 

 specified physically — and taking a as the radius and I the length of the rod, we 



must have (i.) the ratio ^ : L small ; (ii.) the ratio / ^'^ : I small. For experi- 



mental purposes the first ratio should not exceed j^^, but the second need only 

 be smaller than about \. Illustrations of the neglect of these conditions are 

 drawn from the experiments of Despretz and of Wiedemann and Franz. Numerical 

 values of L are given for a range of substances, and the limits between which the 

 Fourier result is applicable are pointed out. A solution having a higher degree 

 of approximation than the Fourier result is then derived from the Bessel function 

 solution, viz. 



'^ Liouville's Journal dc Math. (2), t. xix. 1874, p. ?,a (§§ 5-8). References 

 to Kroneckcr's methods of reduction and to otliermetliods will be found in my paper 

 already quoted. 



- The paper is published in the Proceedings of the London Mathematical Society, 

 vol. xxxiv. 



