On the Connexion between two Physical Magnitudes, 441 



seems to have been independent of Legendre's ; and I had my- 

 self arrived at the same result before I had seen what these 

 mathematicians had written on the subject. See pages ix and 

 427 of my ' History of the Calculus of Variations/ 



The problem as discussed by Professor Challis is free from 

 this condition ; it maybe enunciated thus : — required to connect 

 the ends of a fixed straight line by a curve of given length so 

 that the area bounded by the curve and the straight line may be 

 a maximum. The result is well known, namely that the curve 

 must be an arc of a circle. The given length must, of course, 

 be greater than the length of the fixed straight line ; Professor 

 Challis by a misprint has less instead of greater. The problem 

 as thus enunciated is one of the oldest and most familiar in the 

 subject; and I believe there has never been any doubt or diffi- 

 culty as to the result, which may be obtained by various unex- 

 ceptionable methods ; I have indicated one of these methods at 

 page 69 of my ' History/ 



I do not accept the results at which Professor Challis arrives 

 with respect to the other two problems he discusses; but I have 

 not leisure to enter into details. I have, I believe, fully solved 

 these problems in an Essay which will be published in the course 

 of the present month. 



I. ToDHUNTER. 



St. John's College, Cambridge, 

 November 6, 1871. 



LVII. On a Correction sometimes required in Curves professing to 

 represent the Connexion between two Physical Magnitudes. By 

 the Hon. J. W. Strutt, M.A. 



THE nature of the correction which is the subject of the 

 present paper, and of not infrequent application in ex- 

 perimental inquiry, will be best understood from an example, as 

 it is a little difficult to state with full generality. Suppose that 

 our object is to determine the distribution of heat in the spec- 

 trum of the sun or any other source of light. A line thermo- 

 pile would be placed in the path of the light, and the deflection 

 of the galvanometer noted for a series of positions. But the 

 observations obtained in this way are not sharp — that is, they 

 do not correspond to definite values of the wave-length or refrac- 

 tive index. In the first place, the spectrum cannot be absolutely 

 pure; at each point there is a certain admixture of neighbouring 

 rays. Further, even if the spectrum were pure, it would still be 

 impossible to operate with a mathematical line of it ; so that 

 the result, instead of belonging to a simple definite value of the 



* Communicated by the Author. 



