V.—On Orthogonal Isothermal Surfaces. Part I. By Professor Tarr. 
[Read January 2d, 1866.—Revised and Improved, January 15th, 1872.] 
The following pages contain, in a comparatively compact form, part of the 
substance of a voluminous paper read to the Society six years ago. Ofthat paper, 
which employed ordinary analysis alone, only a few pages had been put in type 
when I succeeded in overcoming a formidable difficulty which had presented 
itself in my quaternion treatment of the subject. I therefore withdrew the 
paper in order that it might have the benefit of the simplification which 
quaternions always give; but it is only of late that I have found time to com- 
plete part of the translation into the new language. From the circumstances 
under which the paper has thus been produced, 7, 7, 4 come forward with undue 
prominence, a thing to be regarded (in HAmILTon’s words) “as an inelegance 
and imperfection in this calculus, or rather in the state to which it has hitherto 
been unfolded.” Immense as is the simplification already attained, it is clear 
that in many places still more is attainable. But I have not postponed my 
paper till it should receive this final polish, partly because the time I can devote 
to such inquiries is extremely limited, and partly because I think that several 
of the results obtained, and of the modes of obtaining them, are new and remark- 
able. Besides, a question of this order of difficulty is admirably adapted to 
show in what respects quaternion methods require improvement. There must 
be some simple mode of deducing (13) and (21) below from (7) without the 
explicit use of 2, 7, #, but I have not yet been fortunate enough to discover it. 
I append to this introduction, for comparison, a few extracts from the paper 
in its original form. 


a. Let dé\2 dé\? dé\? i 
=) a 3 “F fa be written €, 
dé dn , dE dn , dE dy 
dadx  dydy dzedz ~ ; DE 7). 
and 
dE dE a 
Eee $ . : Atém, ¢) 
Bi elit Ms 
de? dy’ da 
a dt at 
dex’ dy’ dz 

VOL. XXVII. PART I. 2H 

