My dear Sir,

Your letter reached me just as I was leaving Cambridge, I had consequently no time to attend to your geometrico-geological Problem.¹ Since I returned however, a day or two ago, I have re-examined it. It is not possible I find by any such modification as you suggest, to make the results accord with your observations on the Northernmost side of granitic axis. The least inclination that could possibly be given to the laminated beds by the anticlinal elevation would be 26^o on the above-mentioned side of the ridge. To produce this result, the dip which must be given by the anticlinal elevation itself (of course in a direction perpendicular to the anticlinal ridge) must equal nearly 70^o, and the strike of your laminated beds would then be perpendicular and therefore their dip parallel to the anticlinal ridge.

By varying the dip of the anticlinal ridge we can vary the dip of the laminated beds from 26^o to any other value up to 90, but then there would be a determinate corresponding position to the line of strike. Thus if we take the least value of the dip (26^o) the direction of the dip would be a long way from the North, and if we take our condition such, that the direction of that dip shall be to the North, then the dip will be much greater than your observed dip, on the North side of the anticlinal line.

As far as this goes, it would prove, or at least would render extremely probable that the lamination was produced after the protrusion of the granitic ridge. By means of another hypothesis however, you might obtain a nearer approximation to your observed results. It is this; the geometrical crest of the anticlinal ridge, instead of being taken horizontal must be supposed to ascend in going towards the North-west, in which case the direction of the dip, produced by it instead of being perpendicular to the anticlinal line will incline more towards the East on the Northernmost side of the anticlinal line, with the corresponding change in the direction of the dip in the Southernmost side of the line. By a proper combination of this new element with the anticlinal dip on the North side of the ridge, we can satisfy the two conditions that the dip of the laminated beds shall have an assigned value, and its direction should coincide with the given direction.

This new hypothesis of a deviation from horizontality in the geometrical ridge of the anticlinal line, is frequently true, towards the extremities of anticlinal lines, as well as in the other parts where there may have been certain irregularities in the elevation. This deviation however, can never exceed a few degrees, if it be continued thro’ any considerable distance. Now if we take 12^o as a mean of your observed dips of the laminated beds on the Northernmost side of the anticlinal line it would require, that the deviation from horizontality just mentioned should be more than 14^o probably not less than 20 in order to satisfy your observations. This amount I conceive to be utterly inadmissible so that I do not see how you can by any admissible hypothesis, account for your observed phenomena in the way you have suggested.

I have used the term geometrical ridge or crest; it may be considered as the line in which the two parts of the same bed respectively on opposite sides of the anticlinal line would meet if produced, assuming as an approximative case, that these two portions of the bed tho’ inclined to the horizon are still plane or flat. The additional hypothesis above mentioned complicates the problem considerably especially in obtaining numerical results. I have not therefore worked out an example with any assumed numerical data. If however you think the additional hypothesis admissable within certain assigned limits, I will work you out an example which will give the best approximation I can make to your observations.

The problem thus generalised is one of great geometrical complexity; but still with three or four pieces of pasteboard, I could in ten minutes interview give you a distinct conception of the problem. The device I before suggested to you is only calculated to give a somewhat rough conception of the problem, and I am not sure on looking at it again whether the numerical values of the angles there given, would make the case very approximate to that which you first proposed to me I should not chose it as the means of explaining the problem generally to any one uninitiated into the mysteries of geometry. The problem is an important one in geological elevations, and occured to myself a considerable time ago, tho’ I have had no occasion to make any exposition of it till your application to me.