464 Mr. S. U. Pickering on the Recognition of 



whether the modification introduced by Prof. Pucker's fourth 

 term bears a sufficiently small proportion to the unmodified 

 part to be correctly designated as a " hump " or not ; but the 

 illustration of converting a circle into an ellipse, which he 

 quotes in justification of the excrescence which he placed on 

 his approximate curve, appears to me to be a singularly 

 unfortunate one, for the ellipse is par excellence the simplest 

 and most natural modification of a circle, and, as such, con- 

 tinually presents itself in natural phenomena. Even if this 

 w T ere not the case, the alteration of one constant in an equation 

 (from 1 to 1—x*) is hardly comparable with the introduction 

 of an entirely new arbitrary function with two arbitrary 

 constants. 



As Prof. Pucker so strenuously defends the legitimacy of 

 his synthesizing his equation, I fail to see why he should 

 object to my analyzing it back again into its original com- 

 ponents, and thus obtaining suggestions of two of the breaks 

 by the very means which was supposed to obliterate them : 

 still less do I appreciate his clinching argument against 

 the validity of these suggestions by showing that analysis in 

 another direction is possible, which, while it fails in suggest- 

 ing one of these two breaks, brings into prominence another 

 one, the only one which I had thought his equation had 

 really obliterated. 



Nor do I see why because I informed Prof. Lodge, in 1889, 

 that I had not then used empirical equations for detecting 

 discontinuities, and did not place much faith in such a 

 method, I should, now that a critic uses empirical equations 

 against me, refrain from examining what evidence they afford 

 for or against these discontinuities. I am merely meeting my 

 opponent on his own ground by doing so. 



(5) Prof. Pucker still considers that his equation was ex- 

 tended sufficiently far beyond the first and fourth of the 

 breaks in question to justify him in saying that it bridged 

 them over ; if, however, instead of taking the average length 

 of the sections of which the whole figure was composed, he 

 had taken the actual lengths of those sections over which the 

 ends of his curve projected, he would have found that the one 

 end projected over rather less than ^ of the next section, the 

 other over rather less than 4- ; distances which, I think, are 

 very insufficient, especially as in the former case the sign of 

 the difference between the observed and calculated values for 

 the last included point is the same as that for the neighbour- 

 ing points beyond it, to which his curve is evidently inapplic- 

 able, showing that the divergence has already begun before 

 this last included point has been reached. 



