Exchange two stored values while preserving all others. For an array, a temporary value implements x := a[i]; a[i] := a[j]; a[j] := x. For distinct indices this uses two element reads and two element writes. The operation is an involution, and its permutation is a transposition.

A right array rotation by $k$ sends the entry originally at offset $j$ to offset $(j+k)\mathrm{mod}n$ for nonempty length $n$. Equivalently it changes adjacent blocks $AB$ into $BA$ when $|B|=k$. Identity rotations include $k=0$ and $k=n$.

Three reversals implement the operation: $AB\to A^{R}B\to A^R{B}^R\to BA$. Alternatively repeatedly use a block swap to put the smaller block in its final position and recurse on the remaining unresolved segment. Replacing that tail recursion by parameter updates gives constant auxiliary storage. If $S(a,b)$ counts swaps for blocks of lengths $a,b>0$, then $S(a,a)=a$ and $S(a,b)=\min (a,b)+S(\min (a,b),|a-b|)$ for unequal lengths. The Euclidean algorithm shows by induction that $S(a,b)=a+b-\gcd (a,b)$, so a right rotation uses $n-\gcd (n,k)$ element swaps for $0<k<n$.

A contiguous portion of an array, or a rectangular restriction to contiguous coordinate ranges for a multidimensional array.

Reverse the order of entries in an array segment by exchanging its outermost unprocessed pair and moving inward. This takes linear time and constant auxiliary space. Reversing two adjacent blocks separately and then reversing their union rotates the blocks while restoring each block's internal order.

Institutional teaching resource: anatomy.ttuhscep.edu/schemes/eye.html

Institutional teaching resource: anatomy.ttuhscep.edu/schemes/eye_ans.html

Institutional teaching resource: www.lumen.luc.edu/lumen/meded/grossanatomy/dissector/labs/h_n/orbit/eye1.html

Institutional teaching resource: learning.medicine.wsu.edu/wp-content/uploads/sites/4/2022/12/PDF-Lab-27.pdf

Institutional teaching resource: anatomy.ttuhscep.edu/schemes/eye_vid_G480.html

Institutional teaching resource: anatomy.ttuhscep.edu/anatomytables/muscles_head_neck.html

Institutional teaching resource: anatomy.ttuhscep.edu/nervous_system/eye_tables.html

Institutional teaching resource: anatomy.ttuhscep.edu/atlas/eye_32.html

Contemporary handbook by William Senhouse Kirkes; p. 69 describes two generations of red corpuscles.

Contemporary monograph by William Baly and William Senhouse Kirkes. See p. 132, Development of the Blood, for competing post-embryonic corpuscle-formation theories.

Contemporary chronicle, British and Foreign Medico-Chirurgical Review, 43(85), pp. 237–247. Part I, item 1 reports a marrow finding published in 1868, after the 1867 exam.

Current reference entry: Introduction, Mechanism and Related Testing; metabolic, synthetic and clearance functions of the liver.

Digestive tract correlates; neurophysiology of relaxation, retrograde contraction, retching and expulsion.

Contemporary review of the 1866 book, pp. 431–440; the tumour discussion is at pp. 438–440. The year in this header identifies the reviewed book, not a verified date of the review. Historical disagreement is evidence about terminology, not current clinical authority.