ON THE SEICHES OE LOCH EARN. 385 



A combination of the two last methods can sometimes be employed with advantage 

 when the fixed limnograph is registering a long sinusoidal series. The index limno- 

 oraph can be used to give very accurately the times of two turning-points, and the 

 fixed limnograph used simply to count the intervening oscillations ; but this does not 

 eliminate error due to disturbance of phase. 



Method of Residuation. — In practice on Loch Earn, more particularly in our 

 attempts to determine the positions of the nodes, we were compelled to work with 

 short, large-scale limnograms ; and the seiches were rarely pure. In these cases we 

 resorted very often to a certain way of treating the limnogram, which we came 

 ultimately to call " Residuation." 



Consider a compound seiche, the equation to whose limnogram is 



q_ 9 



?/ = A 1 sin — {t-a 1 ) + A,, sin ~ (t - a 2 ) + . . . (18). 



l l J- 2 



Construct a new curve by slipping the curve (18) a distance r backwards along the 

 £-axis, and from these two curves form a new one by adding the ordinates ; or, what 

 comes to the same thing, derive from (18) a new curve by adding to the ordinate at 

 each point the ordinate of the point whose abscissa is greater by t. 

 The equation of the resulting curve is 



.t A ITT . 2-7T /, , t\ , ~ , 7TT . 2w I , , t\ , /l n\ 



V = 2A 1 cos —sin _^- ai + -J + 2A 2 cos^ sin - [t-a 2 + -)+ . (19). 



The derived curve, or residual with respect to r, contains in general all the harmonic 

 oscillations of the original. The phase of each compound is retarded by the same 

 time, ^ T ; and the amplitudes are altered in different proportions, viz., 2 cos (ttt/Tj) : 1, 

 2 cos (ttt/T 2 ) : 1, etc. 



Suppose, in particular, that we put t = ^T-^ ; then the first component disappears 

 altogether, and we get a curve, whose equation is 



7 ?1 = 2A 2 cosgi^-a 1 + i)+etc. . . . (20). 



This last curve we call the residual of the limnogram with respect to the seiche of 

 period T v 



To show how this may be used in practice, suppose we have a short, large-scale 

 limnogram, the principal or only components in which are the uninodal seiche (T x ) and 

 the binodal (T 2 ). In general, one, say the uninodal, will predominate ; the other may 

 be scarcely perceptible at first sight. Selecting two turning-points,* we divide the 

 difference between the corresponding points by the number of intervening major 

 oscillations. The result will be an approximation to T 1} say T\ ; but generally only 

 an approximation, because the turning-points are displaced by the other seiches, 

 mainly by the binodal. 



* Points of symmetry, of course, if such are available. 

 TRANS. ROY. SOC. EDIN, VOL. XLV. PART II. (NO. 14). 54 



