The quote in Mariano's answer is from the introduction to Sylvester's paper. Typical of Sylvester's mathematical papers, he used so many nonstandard terms in that paper that he appended a five-page "Glossary of new or unusual Terms, or of Terms used in a new or unusual sense in the preceding Memoir". There he lists:
Inertia. -- The unchangeable number of integers in the excess of positive over negative signs which adheres to a quadratic form expressed as the sum of positive and negative squares, notwithstanding any real linear transformations impressed upon such form.
Sylvester did similarly for many mathematical terms, i.e. coined them or used them in a "new or unusual ways" mathematically. You can find many such examples in Jeff Miller's Earliest Known Uses of Some of the Words of mathematics, including: allotrious factor, anallagmatic, Bezoutiant, catalecticant, combinant covariant cumulant cyclotomy, cyclotomic, dialytic, discriminant, Hessian, invariant, isomorphic, Jacobian, latent, law of intertia of quadratic forms, matrix, minor, nullity, plagiograph, quintic, Schur complement, sequence, syzygy, totient, tree, umbral calculus, umbral notation, universal algebra, x/y/z-coordinate, zero matrix, zetaic multiplication. Please see each entry for Sylvester's role - some are major, others are minor.
Apparently Sylvester's penchant for colorfully naming mathematical objects arose from his love of language and poetry. Indeed, Karen Parshall wrote:
Sylvester's love of poetry and language manifested itself in notable ways even in his mathematical writings. His mastery of French, German, Italian, and Greek was often reflected in the mathematical neologisms - like "meicatecticizant" and "tamisage" - for which he gained a certain notoriety. Moreover, literary illusions, poetic quotations, and unfettered hyperbole spiced his published papers and lectures.
Sylvester wrote about such:
Perhaps I may without immodesty lay claim to the appellation of Mathematical Adam, as I believe that I have given more names (passed into general circulation) of the creatures of mathematical reason than all the other mathematicians of the age combined.-- James Joseph Sylvester, Nature 37 (1888), p. 152.
You can find a short Sylvester biography here.
If $(u,v)$ are isothermal coordinates, then $u$ and $v$ are harmonic functions with respect to the Laplace-Beltrami operator on your Riemannian manifold, that is $\Delta u=\Delta v=0$.
Now, the equation $\Delta f=0$ characterizes the stationary states for the heat equation. The level curves for a harmonic function are therefore the isothermal curves for some heat distribution.
It follows that if $(u,v)$ is an isothermal system of coordinates, then the level curves, i.e., the coordinate lines, are isothermal curves.
Best Answer
I have never heard this referred to as "King's Property" or "King's Rule". And there is almost nothing (except this thread) in Google. There is this video, maybe your answer is there. (Video can be viewed only by paid students) Maybe the term "King's Property" is unique to that video.
The video is from an educational company (?) called Nucleon, located in Rajasthan, in northern India. "Rajasthan" means "Land of Kings".
As pointed out in a comment, the "King's Property" has been mentioned in math.se before. The oldest one seems to be Trigonometric definite integration . That post is from July, 2017. The post is by Tarun Raj Latiyan. Wouldn't it be interesting if Tarun is from India and learned this terminology from the Nucleon video?