162 REPORT— 1898. 



Take a millimetre scale and a sheet of squared paper ruled in millimetres. 

 Mark off OP and OQ at right angles, these lines being taken to represent 

 the magnitude of the two effects. PQ is the length required, being the 

 root of the sum of the squares of the two separate effects. 



12. It must next be shown how to adapt the homologue for any 

 values of E, and L. This is done by means of scale lines. 



A scale line is a line drawn on the chart and graduated by the 

 logaritlnnic rulings of the paper. It is so placed as to read directly the 

 particular value of the quantity to which it refers, such reading being 

 the indication of the position of the homologue on the paper, and is the 

 magnitude of the quantity in the equation which the homologue represents 

 in its present position. 



13. Scale Line for R. — If H, n, and I be all multij^lied by the same 

 quantity the equation is unaltered ; thus when R becomes mR the homo- 

 logue is to be moved a distance log m to the right, and log in upwards. 

 The direction of translation is along the self-induction line, which could 

 be used as a scale line ; in fact any line not parallel to the axis of n 

 could in this case be used as a scale line, the numbers of the graduations 

 being identical with the impedance. In the chart a line at 45° to the 

 axes is drawn and marked ' Scale Line for Resistance.' The direction of 

 motion of the homologue is, in this case, parallel to the scale line. 



Scale Line for L. — If L and n vary inversely the value of I is 

 unchanged. If L become mL the homologue must be moved to the left 

 through a distance log m. The scale line must then be parallel to the 

 axis of impedance ; when the homologue moves any distance to the left 

 the point of intersection of the self-induction and scale lines will then 

 move upwards by the same amount. To avoid specially graduating the 

 scale line it is drawn through a point on the self-induction line where 

 the impedance reading has the same significant figures as the value of L, 

 for which the self-induction line is drawn. The scale line on the chart is 

 drawn through the point of intersection of 1=2 with the self-induction 

 line. 



14. To find the proper position of the homologue for any value of R 

 and L, move the whole curve with its attendant resistance and self-induc- 

 tion lines by a motion of pure translation until the resistance and self- 

 induction lines cut their respective scale lines at the appropriate readings 

 for R and L. 



Discussion op a Non-translatant. 



15. Even in the case of some curves which are not translatants the 

 labour of plotting their equations may be greatly reduced by the use of 

 this method. To illustrate this let us take an equation which does not 

 represent a translatant curve. 



Professor Greenhill has given the theory of waves on a frozen sea in 

 his article ' Wave Motion in Hydrodynamics.' ' By retaining a term for 

 the density of the solid in this investigation we have the velocity of pro- 

 pagation and the wave-length connected by the equation 



r/t s 2tre\ 



H^-'^^/H'^-'- 



• Avterican Journal of Mathematics, vol. is. No. 1. 



