iy02.] MACKE.VZIE— EQUATIONS OF HEAT PKOPAGATIOX. 199 



of such Strips is not the exact area required ; f yt/x is the limit 



toward which the sum tends as (/x tends to zero, and we know from 

 the familiar example of Fourier's series how the value can change 

 actually in the limit. It happens in the present case that as ^c is 

 made smaller and smaller, and V^ correspondingly greater and 



greater in order to keep cr constant, in the limit — - — ^ is the 



exact solution for an infinite plane (see under (21) and (22)). So 

 in making the integration above, that is, in finding the limit of the 

 summation, we get necessarily an exact solution because in the limit 

 each term of the solution is exact. Had we approached the limit 

 in some other way than in keeping ^ constant we might have got 

 quite a different result. 



The forms of the curves for (27) are shown on Plates XXVII 

 and XXVIII. Curves £\E^ and £^ are drawn with values of x as 



abscissae for values of the time — ^, -jy- and - respectively ; curves 



J^\ F"^ and F^ are drawn with values of ikt as abscissae for values 

 of the distance j, ^, and 1 respectively. 



Since the current or flow is got from the temperature by a differ- 

 entiation with regard to x, and since equation (27) was got from 

 (20) by an integration with regard to x, it is evident that the 

 curves for the potential or temperature in (20) are the curves for 

 currer.t in the present problem. 



I=-KZ = ^^.e (28) 



dx 



%{r:ktY 



These curves are given on Plates XXIII and XXIV for points to 

 the right of the origin ; the form for points to the left is obvious, 

 since the curves are symmetrical about \.\-\q yz plane. 



Physical Laboratory, Bryn Mawr College. 

 April J, I go 2. 



