JANUAKY 12, 1900.] 



SCIENCE. 



45 



by the equations of Lagrange, and by the 

 equation of Hamilton. Each way has 

 special advantages for particular applica- 

 tions, and together they may be said to 

 condense into the narrow space of a few 

 printed lines the net results of more than 

 twenty centuries of effort in the formula- 

 tion of the phenomena of matter and motion. 



Sacli was the state of mechanical science 

 when the great physical discovery of the 

 century, the law of conservation of energy, 

 was made. To give adequate expression 

 to this law it was only necessary to recur 

 to the Mecanique Analytique, for herein 

 Lagrange had prepared almost all of the 

 needful machinery. So well indeed were 

 the ideas and methods of Lagrange adapted 

 to this purpose that they have not only 

 furnished the points of departure for many 

 of the most important discoveries* of the 

 present half century, but they have also 

 supplied the criteria by means of which me- 

 chanical phenomena in general are most 

 easily and effectively defined and inter- 

 preted. 



Of the special branches of analytical me- 

 chanics which have undergone development 

 during this century, by far the most impor- 

 tant is that known as the theory of the 

 potential function. This function first 

 appeared in mathematical analysis in a 

 memoir of Lagrange in 1777 f as the ex- 

 pression of the perturbative function, or 

 force function. It next appeared in 1782 % 

 in a memoir by Laplace. In this memoir 

 Laplace's equation § appears for the first 

 time, being here expressed in polar coordi- 



* Especially those in the theories of electricity, 

 magnetism, and thermodynamics. 



^ Nouveaux Mhnoires de I'Academie des Sciences et 

 Belles Leitres de Berlin. See also remarks of Heine, 

 Handbuch der Kugelfunctionen, Band II., p. 342. 



X Paris Memoires for 1782, published in 1785. 



iV= 



32 r 



+ ■ 



A»F is called the Laplacian of V. 



= 0. 



nates. In 1787* the same equation ap- 

 pears in the more usual form as expressed 

 by rectangular coordinates. 



Strange as it now seems when viewed by 

 the light of this end of the century, nearly 

 thirty years elapsed before Laplace's equa- 

 tion was generalized. Laplace had found 

 only half of the truth, namely, that which 

 applies to points external to the attracting 

 masses. -|- Poisson discovered the other half 

 in 181 3. J Thus the honors attached to the 

 introduction of this remarkable theoi-em 

 are divided between them, and we now 

 speak of the equation of Laplace and the 

 equation of Poisson, though the equation of 

 Poisson includes that of Laplace. 



Next came the splendid contributions of 

 George Green under the modest title of 

 " An essay on the application of mathe- 

 matical analysis to the theories of electricity 

 and magnetism. "§ It is in this essay that 

 the term ' potential function ' first occurs. 

 Herein also his remarkable theorem in pure 

 mathematics, since universally known as 

 Green's theorem, and probably the most im- 

 portant instrument of investigation in the 

 whole range of mathematical physics, made 

 its appearance. 



We are all now able to understand, in a 

 general way at least, the importance of 

 Green's work, and the progress made since 

 the publication of his essay in 1828. But 

 fully to appreciate his work and subsequent 

 progress, one needs to know the outlook for 

 the mathematico-physical sciences as it ap- 

 peared to Green at this time, and to realize 

 his refined sensitiveness in promulgating 

 his discoveries. 



* Paris Memoires for 1787, published in 1789. 



fThat is, Laplace's equation is /\'^V=(i, while 

 Poisson's is A^ p -|- i-Jcp = 0, V being the potential 

 and p the density at the point {x, y, z) , and k being the 

 gravitation constant. 



% Poisson's equation was derived in a paper pub- 

 lished in Nouwaxi Bulletin * * * Societe Pltilomatique, 

 Paris, Dec, 1813. 



? Nottingham, 1828. 



