172 REPORT—1862. 
at all were so far extended as to embrace the whole period of the cyclical 
changes. 
The simultaneous observation of the irregular changes is limited nearly to 
the same stations. In their case, too, there is the further imperfection, as 
respects the present problem, that the changes observed on “ term-days” 
are for the most part inconsiderable, while those on days of great magnetic 
disturbance have seldom been observed continuously for any considerable 
time at all the stations. 
For the foregoing reasons the Committee are of opinion that the data which 
we at present possess respecting the changes of terrestrial magnetism, whether 
regular or irregular, are not sufficient for the application of Gauss’s theory, 
if, as above assumed, the approximation is to be extended so as to include 
terms of the fourth order (P, to P, inclusive). It is deserving of considera- 
tion, however, whether an inferior degree of approximation may not afford 
some valuable information. The affirmative side of this question has been so 
earnestly advocated by one of the members of the Committee, that it has been 
thought advisable to append his letter on the subject to this Report. 
(Signed by order of the Committee) H. Lioyp. 
Letter from Professor W. Tuomson to Rev. Dr. Lioyp. 
“‘ Roshyen, Strontian, Sept. 24, 1862. 
«‘ My prar Sr,—I am sorry to have been so long prevented from writing 
to you on the subject of the Committee’s Report on the expression of the 
Variations of the Terrestrial Magnetic elements in series of Laplace’s functions. 
“T perfectly agree with the conclusions stated in the draft report of which 
you sent me a proof, so far as they relate to a complete expression of any class 
of variations of the elements, or of any individual variation, by means of 
which its amount in other localities than those of observation could be de- 
termined with any considerable approach to accuracy. But, on the other 
hand, the amount of knowledge from observation, shown in the report to be 
available, would, I believe, be sufficient to allow us to estimate, possibly with 
considerable accuracy, and certainly with a sufficient approach to accuracy 
for highly important application, the first terms in the harmonic (Laplace’s) 
series. I would therefore advise that some such method as the following 
should be adopted. 
«‘ Choosing any particular variation, for instance the diurnal or the secular, 
for which the data from observation are most abundant, find either by trial 
and error, or any other proper algebraic method, an expression by terms of 
the first order (three coefficients for each) for the three elements which most 
nearly represent it. (The method of least squares would give a precise de- 
finition of what would be the most near representation, on this principle ; but 
ruder and quicker methods might suffice in first trials.) Then, judging by 
the results, try similarly for expressions in series of two terms (3+ 5, or eight 
coefficients in all, in each expression). After trials of this kind it would be 
easy to judge within what limits may be the probable errors of the estimated 
first terms from the true first terms, and possibly even to arrive at some 
probable knowledge regarding the true second terms of the harmonic ex- 
pressions. 
«« A very moderate degree of success in such operations as these would 
allow us to decide whether the origin (magnetic or electrodynamic) of the 
variation is within the earth’s surface or outside. 
igs 
